WEBVTT 00:00:00.000 --> 00:00:01.984 align:middle line:90% [SQUEAKING] 00:00:01.984 --> 00:00:03.472 align:middle line:90% [RUSTLING] 00:00:03.472 --> 00:00:04.960 align:middle line:90% [CLICKING] 00:00:04.960 --> 00:00:11.420 align:middle line:90% 00:00:11.420 --> 00:00:13.220 align:middle line:84% PAIGE BRIGHT: So today, in case you 00:00:13.220 --> 00:00:15.860 align:middle line:84% were thinking that there wasn't enough theorems on Tuesday, 00:00:15.860 --> 00:00:17.277 align:middle line:84% we're going to prove a lot of them 00:00:17.277 --> 00:00:19.730 align:middle line:84% today on the general metrics based theory. 00:00:19.730 --> 00:00:22.520 align:middle line:84% I like this notion of a theory of the spaces. 00:00:22.520 --> 00:00:26.960 align:middle line:84% This is a notation that Dr. Casey Rodriguez taught me, 00:00:26.960 --> 00:00:28.610 align:middle line:84% loosely speaking, where you're just 00:00:28.610 --> 00:00:31.040 align:middle line:84% trying to encapsulate what are all the things we can say 00:00:31.040 --> 00:00:33.560 align:middle line:84% about the most fundamental objects in the space 00:00:33.560 --> 00:00:34.850 align:middle line:90% that you're considering? 00:00:34.850 --> 00:00:38.540 align:middle line:84% And essentially, a number of the theorems that we'll prove 00:00:38.540 --> 00:00:41.543 align:middle line:84% are essentially the same as what we would picture in Euclidean 00:00:41.543 --> 00:00:43.460 align:middle line:84% space, but we have to be slightly more careful 00:00:43.460 --> 00:00:44.690 align:middle line:90% sometimes. 00:00:44.690 --> 00:00:46.880 align:middle line:84% And all of this theory is going to be building up 00:00:46.880 --> 00:00:48.765 align:middle line:84% to understanding something that you probably 00:00:48.765 --> 00:00:50.390 align:middle line:84% haven't seen before if you haven't done 00:00:50.390 --> 00:00:53.540 align:middle line:90% 100B, known as compact sets. 00:00:53.540 --> 00:00:57.350 align:middle line:84% So we're going to be building up to talking about compact sets. 00:00:57.350 --> 00:00:59.228 align:middle line:84% But first, we have to understand what 00:00:59.228 --> 00:01:00.770 align:middle line:84% are the fundamental tools in our tool 00:01:00.770 --> 00:01:03.037 align:middle line:90% belt with metrics based theory? 00:01:03.037 --> 00:01:04.370 align:middle line:90% And I do want to note one thing. 00:01:04.370 --> 00:01:05.900 align:middle line:84% Even though last time, I said we can 00:01:05.900 --> 00:01:08.330 align:middle line:84% put a metric on every set, which is true, 00:01:08.330 --> 00:01:12.710 align:middle line:84% not every set has a nice metric on it. 00:01:12.710 --> 00:01:15.370 align:middle line:84% Have any of y'all taken topology before? 00:01:15.370 --> 00:01:16.120 align:middle line:90% No worries if not. 00:01:16.120 --> 00:01:18.700 align:middle line:84% It's a class that usually comes after real analysis, 00:01:18.700 --> 00:01:23.500 align:middle line:84% but in topology talk, not every topological space 00:01:23.500 --> 00:01:27.700 align:middle line:84% is metrizable-- i.e. there's not necessarily a metric on it that 00:01:27.700 --> 00:01:28.900 align:middle line:90% will make the topology nice. 00:01:28.900 --> 00:01:31.378 align:middle line:84% And we'll talk about what topology is a little bit today. 00:01:31.378 --> 00:01:33.670 align:middle line:84% So in case you're interested in what that statement is, 00:01:33.670 --> 00:01:35.290 align:middle line:84% we'll come to it, but the basic idea 00:01:35.290 --> 00:01:38.770 align:middle line:84% is even though we can put a metric on every set, 00:01:38.770 --> 00:01:41.720 align:middle line:84% sometimes the trivial one is not that interesting. 00:01:41.720 --> 00:01:44.300 align:middle line:90% So all right. 00:01:44.300 --> 00:01:48.045 align:middle line:84% Let's go ahead and jump into the metrics based theory for today. 00:01:48.045 --> 00:01:49.670 align:middle line:84% So first off, we're going to be talking 00:01:49.670 --> 00:01:52.100 align:middle line:84% about convergent sequences, just like we did in RN. 00:01:52.100 --> 00:02:00.320 align:middle line:84% So throughout this, let xn and yn 00:02:00.320 --> 00:02:02.240 align:middle line:84% be the sequences we're considering. 00:02:02.240 --> 00:02:05.340 align:middle line:90% 00:02:05.340 --> 00:02:16.200 align:middle line:84% In the first statement theorem, suppose xn converges to x. 00:02:16.200 --> 00:02:19.410 align:middle line:84% What should we expect about the point that it converges to? 00:02:19.410 --> 00:02:22.290 align:middle line:84% Well, in Euclidean space, we knew that this limit point 00:02:22.290 --> 00:02:23.100 align:middle line:90% had to be unique. 00:02:23.100 --> 00:02:25.800 align:middle line:84% And in fact, we can prove that this must be the case. 00:02:25.800 --> 00:02:28.410 align:middle line:90% 00:02:28.410 --> 00:02:32.675 align:middle line:90% So then x is unique. 00:02:32.675 --> 00:02:36.700 align:middle line:90% 00:02:36.700 --> 00:02:40.450 align:middle line:84% To prove a statement like this, we choose a different point y 00:02:40.450 --> 00:02:42.080 align:middle line:90% in our set to consider. 00:02:42.080 --> 00:02:44.590 align:middle line:84% So what we're going to show now is that convergent sequences 00:02:44.590 --> 00:02:46.850 align:middle line:90% have unique limit points. 00:02:46.850 --> 00:02:49.350 align:middle line:90% So proof. 00:02:49.350 --> 00:02:55.740 align:middle line:84% Suppose there exists a y in your metric space x-- 00:02:55.740 --> 00:02:59.850 align:middle line:84% or I should say sequences in your metric space xd, 00:02:59.850 --> 00:03:03.270 align:middle line:90% so metric x with metric d. 00:03:03.270 --> 00:03:04.770 align:middle line:84% And suppose that there exists a y 00:03:04.770 --> 00:03:12.180 align:middle line:84% and x such that xn converges to y. 00:03:12.180 --> 00:03:15.840 align:middle line:84% Our end goal is to show then that x must be equal to y, 00:03:15.840 --> 00:03:17.220 align:middle line:90% in fact. 00:03:17.220 --> 00:03:21.720 align:middle line:84% Now, to do this, what do we know about equality for points? 00:03:21.720 --> 00:03:25.020 align:middle line:84% Well, by the axioms of what we're defining a metric space 00:03:25.020 --> 00:03:27.550 align:middle line:84% to be, two points are going to be the same 00:03:27.550 --> 00:03:29.160 align:middle line:90% if the distance is precisely 0. 00:03:29.160 --> 00:03:31.347 align:middle line:84% It's an if and only if statement. 00:03:31.347 --> 00:03:32.680 align:middle line:90% So that's what we want to prove. 00:03:32.680 --> 00:03:35.880 align:middle line:84% We want to prove that the distance between x and y is 0. 00:03:35.880 --> 00:03:37.870 align:middle line:90% So how can we do that? 00:03:37.870 --> 00:03:40.710 align:middle line:84% Well, firstly, let's write down what 00:03:40.710 --> 00:03:43.170 align:middle line:84% the definitions of them converging to x and y 00:03:43.170 --> 00:03:46.110 align:middle line:90% is respectively. 00:03:46.110 --> 00:03:48.660 align:middle line:90% So xn converges to x. 00:03:48.660 --> 00:03:51.930 align:middle line:84% Means that for all epsilon bigger than 0, 00:03:51.930 --> 00:03:55.230 align:middle line:84% there exists an n in the natural numbers 00:03:55.230 --> 00:03:59.400 align:middle line:84% such that for all n bigger than or equal to n-- 00:03:59.400 --> 00:04:01.680 align:middle line:90% and I'll call this n1-- 00:04:01.680 --> 00:04:06.645 align:middle line:84% we have that the distance from xn to x is less than epsilon. 00:04:06.645 --> 00:04:08.520 align:middle line:84% This is the definition of convergent sequence 00:04:08.520 --> 00:04:10.800 align:middle line:90% that we're dealing with. 00:04:10.800 --> 00:04:15.720 align:middle line:84% And similarly, xn converges to y if there exists 00:04:15.720 --> 00:04:20.310 align:middle line:84% an n2 in the natural numbers for this epsilon such 00:04:20.310 --> 00:04:26.550 align:middle line:84% that for n bigger than or equal to n2, 00:04:26.550 --> 00:04:31.810 align:middle line:84% the distance from xn to y is less than epsilon. 00:04:31.810 --> 00:04:33.240 align:middle line:90% So now, what do we want to do? 00:04:33.240 --> 00:04:36.510 align:middle line:84% We want to use this to show that the distance between x and y 00:04:36.510 --> 00:04:37.470 align:middle line:90% is small. 00:04:37.470 --> 00:04:39.270 align:middle line:84% And to do this, we will apply the triangle 00:04:39.270 --> 00:04:44.190 align:middle line:84% inequality to understand the distance from x to y. 00:04:44.190 --> 00:04:46.440 align:middle line:84% So by the triangle inequality, this 00:04:46.440 --> 00:04:49.110 align:middle line:84% is less than or equal to the distance from xn 00:04:49.110 --> 00:04:53.703 align:middle line:84% to x plus the distance from xn to y, 00:04:53.703 --> 00:04:55.620 align:middle line:84% where here I'm applying symmetry so that I can 00:04:55.620 --> 00:04:58.380 align:middle line:90% move the terms around, right? 00:04:58.380 --> 00:05:02.610 align:middle line:84% And we know that for all epsilon bigger than 0, 00:05:02.610 --> 00:05:04.500 align:middle line:84% there exists an n-- let's say this 00:05:04.500 --> 00:05:12.760 align:middle line:84% is equal to the maximum of n1 and n2 in the natural numbers 00:05:12.760 --> 00:05:16.240 align:middle line:84% such that both of these terms are now less than epsilon. 00:05:16.240 --> 00:05:17.230 align:middle line:90% In fact, you know what? 00:05:17.230 --> 00:05:20.920 align:middle line:84% To make it nicer, I'll make this epsilon over 2 both spots. 00:05:20.920 --> 00:05:23.710 align:middle line:84% So then this is less than epsilon over 2 00:05:23.710 --> 00:05:28.460 align:middle line:84% plus epsilon over 2, which is equal to epsilon. 00:05:28.460 --> 00:05:33.190 align:middle line:84% Now, does this imply immediately that the distance between them 00:05:33.190 --> 00:05:34.108 align:middle line:90% is 0? 00:05:34.108 --> 00:05:35.650 align:middle line:84% Essentially, we have to note one more 00:05:35.650 --> 00:05:39.830 align:middle line:84% thing, which is that distances are positive definite. 00:05:39.830 --> 00:05:42.190 align:middle line:84% So this is going to be bigger than or equal to 0. 00:05:42.190 --> 00:05:46.210 align:middle line:84% So what this tells you is that because the distance between x 00:05:46.210 --> 00:05:50.500 align:middle line:84% and y gets arbitrarily small and arbitrarily 00:05:50.500 --> 00:05:52.720 align:middle line:84% small and close to 0, it has to be 0. 00:05:52.720 --> 00:05:56.540 align:middle line:84% This is from the real number theory that we already know. 00:05:56.540 --> 00:06:01.030 align:middle line:84% So this implies that the distance from x to y is 0. 00:06:01.030 --> 00:06:04.960 align:middle line:84% And therefore, x is equal to y, which 00:06:04.960 --> 00:06:07.220 align:middle line:90% is what we wanted to prove. 00:06:07.220 --> 00:06:11.005 align:middle line:84% So notice that a lot of this used all of the axioms 00:06:11.005 --> 00:06:12.130 align:middle line:90% that we were talking about. 00:06:12.130 --> 00:06:14.590 align:middle line:84% At one point, we use symmetry, which wasn't too important, 00:06:14.590 --> 00:06:16.660 align:middle line:84% but was useful to note so that we 00:06:16.660 --> 00:06:19.930 align:middle line:84% can swap xn and x around and apply the triangle inequality. 00:06:19.930 --> 00:06:21.550 align:middle line:84% We applied the triangle inequality, 00:06:21.550 --> 00:06:24.520 align:middle line:84% and we had to use positive definiteness. 00:06:24.520 --> 00:06:26.920 align:middle line:84% So these are the bare bones of what we really 00:06:26.920 --> 00:06:29.740 align:middle line:90% need in the theory. 00:06:29.740 --> 00:06:32.380 align:middle line:90% All right. 00:06:32.380 --> 00:06:34.270 align:middle line:84% So now what we're going to do is show 00:06:34.270 --> 00:06:36.460 align:middle line:84% that not only do limit points exist nicely, 00:06:36.460 --> 00:06:40.750 align:middle line:84% but the distances between limit points act nicely. 00:06:40.750 --> 00:06:45.250 align:middle line:84% So to state that more clearly, theorem-- 00:06:45.250 --> 00:06:53.310 align:middle line:84% let y be an x and xn converge to x. 00:06:53.310 --> 00:06:58.830 align:middle line:84% Then my claim is that the distance from xn to y 00:06:58.830 --> 00:07:03.730 align:middle line:84% will converge from the distance from x to y, 00:07:03.730 --> 00:07:05.260 align:middle line:90% which makes relative sense. 00:07:05.260 --> 00:07:08.050 align:middle line:84% But notice here the statement is slightly different. 00:07:08.050 --> 00:07:10.630 align:middle line:84% Before, we were purely dealing with sequences 00:07:10.630 --> 00:07:13.080 align:middle line:90% of a metric space. 00:07:13.080 --> 00:07:16.460 align:middle line:84% And so we had to deal with the metric d itself. 00:07:16.460 --> 00:07:19.670 align:middle line:84% But when we're saying that this converges to d xy, 00:07:19.670 --> 00:07:22.280 align:middle line:84% what space am I considering these distances in? 00:07:22.280 --> 00:07:25.790 align:middle line:90% 00:07:25.790 --> 00:07:26.290 align:middle line:90% Anyone? 00:07:26.290 --> 00:07:29.460 align:middle line:90% 00:07:29.460 --> 00:07:31.380 align:middle line:90% No worries. 00:07:31.380 --> 00:07:33.120 align:middle line:90% We're looking at it in r, right? 00:07:33.120 --> 00:07:35.920 align:middle line:84% Because for every single n, this is just a real number. 00:07:35.920 --> 00:07:37.680 align:middle line:84% And so when I say the distance from xn 00:07:37.680 --> 00:07:39.870 align:middle line:84% to y converges to the distance from x to y, 00:07:39.870 --> 00:07:43.110 align:middle line:84% I mean that this happens in Euclidean space. 00:07:43.110 --> 00:07:45.210 align:middle line:90% And in fact, you can likely-- 00:07:45.210 --> 00:07:46.638 align:middle line:84% I don't want to state explicitly, 00:07:46.638 --> 00:07:48.180 align:middle line:84% but I'm pretty certain can prove this 00:07:48.180 --> 00:07:52.770 align:middle line:84% as well in a general setting, so between two metric spaces. 00:07:52.770 --> 00:07:56.962 align:middle line:90% OK, so let's prove this. 00:07:56.962 --> 00:08:03.210 align:middle line:84% To do this, ultimately, we want to show 00:08:03.210 --> 00:08:06.090 align:middle line:84% that for all epsilon bigger than 0, 00:08:06.090 --> 00:08:09.060 align:middle line:84% there exists an n in the natural numbers 00:08:09.060 --> 00:08:14.490 align:middle line:84% such that the distance from xn to y minus the distance 00:08:14.490 --> 00:08:19.540 align:middle line:84% from x to y is less than epsilon in absolute values. 00:08:19.540 --> 00:08:21.330 align:middle line:84% And to do so, we can just find an upper 00:08:21.330 --> 00:08:27.080 align:middle line:84% and lower bound on the distance from xn to y. 00:08:27.080 --> 00:08:28.280 align:middle line:90% So let's do that. 00:08:28.280 --> 00:08:30.860 align:middle line:90% The first direction is nice. 00:08:30.860 --> 00:08:33.799 align:middle line:84% The distance from xn to y is less than 00:08:33.799 --> 00:08:36.049 align:middle line:84% or equal to, by the triangle inequality, the distance 00:08:36.049 --> 00:08:41.179 align:middle line:84% from xn to x plus the distance from x to y. 00:08:41.179 --> 00:08:43.860 align:middle line:84% And we can make this term arbitrarily small. 00:08:43.860 --> 00:08:46.700 align:middle line:84% So I'm just going to choose the same n 00:08:46.700 --> 00:08:49.200 align:middle line:84% to be the one for this convergent sequence. 00:08:49.200 --> 00:08:54.865 align:middle line:84% So this is less than epsilon plus the distance from x to y. 00:08:54.865 --> 00:08:56.990 align:middle line:84% So this gives us the upper bound that we're wanting 00:08:56.990 --> 00:08:58.823 align:middle line:90% on the distance from xn to y. 00:08:58.823 --> 00:08:59.990 align:middle line:90% Let's prove the lower bound. 00:08:59.990 --> 00:09:01.490 align:middle line:84% And the lower bound is very similar, 00:09:01.490 --> 00:09:04.010 align:middle line:84% but it's just a slightly different manipulation, where 00:09:04.010 --> 00:09:05.570 align:middle line:90% what we're then going to do-- 00:09:05.570 --> 00:09:09.670 align:middle line:90% so the distance from xn to y-- 00:09:09.670 --> 00:09:12.220 align:middle line:84% or actually, I'll look at the distance from x to y. 00:09:12.220 --> 00:09:13.970 align:middle line:84% This is less than or equal to the distance 00:09:13.970 --> 00:09:19.970 align:middle line:84% from x to xn plus the distance from x to y. 00:09:19.970 --> 00:09:23.790 align:middle line:84% And then what we can do is subtract these terms. 00:09:23.790 --> 00:09:26.090 align:middle line:84% So specifically, we'll get that the distance from x 00:09:26.090 --> 00:09:35.840 align:middle line:84% to y minus the distance from x to xn is less than 00:09:35.840 --> 00:09:38.780 align:middle line:84% or equal to the distance from x to y. 00:09:38.780 --> 00:09:40.740 align:middle line:84% And we can make this term arbitrarily small. 00:09:40.740 --> 00:09:42.115 align:middle line:84% So this is less than the distance 00:09:42.115 --> 00:09:45.440 align:middle line:90% from x to y minus epsilon. 00:09:45.440 --> 00:09:48.290 align:middle line:84% And notice that this implies the result 00:09:48.290 --> 00:09:53.120 align:middle line:84% because then we have that the distance from xn to y is less-- 00:09:53.120 --> 00:09:55.530 align:middle line:90% oh, shoot. 00:09:55.530 --> 00:09:56.550 align:middle line:90% Did I get this wrong? 00:09:56.550 --> 00:09:58.960 align:middle line:90% 00:09:58.960 --> 00:09:59.460 align:middle line:90% No, no. 00:09:59.460 --> 00:10:00.030 align:middle line:90% OK, yeah. 00:10:00.030 --> 00:10:06.080 align:middle line:84% So then the distance from xn y is less than-- 00:10:06.080 --> 00:10:08.510 align:middle line:84% sorry, this should be the distance from x to y. 00:10:08.510 --> 00:10:09.740 align:middle line:90% There we go. 00:10:09.740 --> 00:10:12.358 align:middle line:90% The distance from xn to-- 00:10:12.358 --> 00:10:13.400 align:middle line:90% let me double-check this. 00:10:13.400 --> 00:10:20.300 align:middle line:84% Distance from xn to y plus xn to y. 00:10:20.300 --> 00:10:21.542 align:middle line:90% There we go. 00:10:21.542 --> 00:10:22.250 align:middle line:90% Sorry about that. 00:10:22.250 --> 00:10:25.480 align:middle line:84% I mixed up what they were converging to. 00:10:25.480 --> 00:10:28.900 align:middle line:84% So what this tells us is that the distance from xn 00:10:28.900 --> 00:10:32.200 align:middle line:84% to y minus the distance from x to y 00:10:32.200 --> 00:10:34.420 align:middle line:90% gets arbitrarily small, right? 00:10:34.420 --> 00:10:36.770 align:middle line:84% I can write that out, if y'all would prefer, 00:10:36.770 --> 00:10:38.020 align:middle line:90% but that's just the next step. 00:10:38.020 --> 00:10:40.760 align:middle line:84% You write it in terms of the absolute values. 00:10:40.760 --> 00:10:41.980 align:middle line:90% OK? 00:10:41.980 --> 00:10:44.840 align:middle line:90% Any questions? 00:10:44.840 --> 00:10:49.760 align:middle line:84% So not only are limit points unique, 00:10:49.760 --> 00:10:51.367 align:middle line:84% but the distances between limit points 00:10:51.367 --> 00:10:53.700 align:middle line:84% are also going to be unique, at least in the real number 00:10:53.700 --> 00:10:56.150 align:middle line:90% sense, which is pretty nice. 00:10:56.150 --> 00:10:57.020 align:middle line:90% I'll remark. 00:10:57.020 --> 00:11:00.510 align:middle line:84% One proposition, which is pretty interesting-- 00:11:00.510 --> 00:11:03.020 align:middle line:84% you can study this for two convergent sequences at once. 00:11:03.020 --> 00:11:08.000 align:middle line:90% 00:11:08.000 --> 00:11:09.290 align:middle line:90% I'll state this in two parts. 00:11:09.290 --> 00:11:15.660 align:middle line:84% Suppose xn converges to x and yn converges to y. 00:11:15.660 --> 00:11:20.960 align:middle line:84% Then the distance from xn to yn converges 00:11:20.960 --> 00:11:24.470 align:middle line:90% to the distance from x to y. 00:11:24.470 --> 00:11:27.230 align:middle line:84% This relatively makes sense, but we can also 00:11:27.230 --> 00:11:29.570 align:middle line:84% state this slightly differently, or a different type 00:11:29.570 --> 00:11:30.360 align:middle line:90% of this theorem. 00:11:30.360 --> 00:11:36.590 align:middle line:90% Suppose xn and yn are Cauchy. 00:11:36.590 --> 00:11:37.935 align:middle line:90% So they're Cauchy sequences. 00:11:37.935 --> 00:11:39.560 align:middle line:84% So they get arbitrarily close together, 00:11:39.560 --> 00:11:44.370 align:middle line:84% but don't necessarily have a limit point. 00:11:44.370 --> 00:11:49.580 align:middle line:84% Then we know that the distance from x to yn converges. 00:11:49.580 --> 00:11:52.800 align:middle line:90% 00:11:52.800 --> 00:11:55.530 align:middle line:84% Notice that I'm not saying what it converges to. 00:11:55.530 --> 00:11:58.200 align:middle line:84% I don't know that Cauchy sequences have limit points 00:11:58.200 --> 00:12:00.455 align:middle line:90% in my metric space. 00:12:00.455 --> 00:12:02.080 align:middle line:84% And we'll get back to that in a moment. 00:12:02.080 --> 00:12:05.130 align:middle line:84% But nonetheless, the sequence itself in the real numbers 00:12:05.130 --> 00:12:06.120 align:middle line:90% will converge. 00:12:06.120 --> 00:12:08.880 align:middle line:84% And that uses the fact that the real numbers is Cauchy 00:12:08.880 --> 00:12:10.347 align:middle line:90% complete. 00:12:10.347 --> 00:12:12.180 align:middle line:84% Now, I'm not going to prove these two facts. 00:12:12.180 --> 00:12:15.127 align:middle line:84% These are facts that are on your problem set, 00:12:15.127 --> 00:12:17.460 align:middle line:84% but I do want to point out-- because this happened a lot 00:12:17.460 --> 00:12:18.103 align:middle line:90% last year-- 00:12:18.103 --> 00:12:20.520 align:middle line:84% you cannot assume that the limit points in the second part 00:12:20.520 --> 00:12:21.600 align:middle line:90% exist. 00:12:21.600 --> 00:12:25.290 align:middle line:84% You can note that this implication implies the top one 00:12:25.290 --> 00:12:29.130 align:middle line:84% by uniqueness because convergent sequences are Cauchy, as we're 00:12:29.130 --> 00:12:31.180 align:middle line:90% going to prove right now. 00:12:31.180 --> 00:12:32.730 align:middle line:84% So if you prove the second one first, 00:12:32.730 --> 00:12:34.325 align:middle line:84% then you can make a small remark. 00:12:34.325 --> 00:12:35.700 align:middle line:84% You have to explain why the limit 00:12:35.700 --> 00:12:38.860 align:middle line:84% point is the distance from x to y, but that's not too bad. 00:12:38.860 --> 00:12:44.282 align:middle line:90% So let's prove that theorem. 00:12:44.282 --> 00:12:45.990 align:middle line:84% Is that the next theorem I wanted to say? 00:12:45.990 --> 00:12:52.690 align:middle line:90% 00:12:52.690 --> 00:12:55.850 align:middle line:90% Yeah, I'll do that one now. 00:12:55.850 --> 00:13:01.700 align:middle line:90% Cauchy sequences-- ope, sorry. 00:13:01.700 --> 00:13:11.510 align:middle line:84% Convergent sequences are Cauchy sequences. 00:13:11.510 --> 00:13:17.220 align:middle line:90% 00:13:17.220 --> 00:13:21.450 align:middle line:84% Suppose xn is the convergent sequence we're considering, 00:13:21.450 --> 00:13:24.550 align:middle line:90% and that xn converges to x. 00:13:24.550 --> 00:13:26.293 align:middle line:90% Then we know that-- 00:13:26.293 --> 00:13:28.210 align:middle line:84% and I'm going to say this over and over again, 00:13:28.210 --> 00:13:29.600 align:middle line:84% just like we do in real analysis, 00:13:29.600 --> 00:13:32.980 align:middle line:84% but at a certain point, it just becomes second-hand nature-- 00:13:32.980 --> 00:13:35.380 align:middle line:84% so therefore, for all epsilon bigger than 0, 00:13:35.380 --> 00:13:38.860 align:middle line:84% there exists n in the natural numbers 00:13:38.860 --> 00:13:47.260 align:middle line:84% such that for all n bigger than or equal to n, 00:13:47.260 --> 00:13:51.800 align:middle line:84% the distance from xn to x is less than epsilon. 00:13:51.800 --> 00:13:53.690 align:middle line:90% So what does this tell us? 00:13:53.690 --> 00:13:56.320 align:middle line:84% Well, what we're ultimately interested in 00:13:56.320 --> 00:14:00.460 align:middle line:84% is the distance from xn to xm, right? 00:14:00.460 --> 00:14:04.840 align:middle line:84% Let m be bigger than or equal to n. 00:14:04.840 --> 00:14:07.240 align:middle line:84% What we're interested in is the distance from xn 00:14:07.240 --> 00:14:10.220 align:middle line:84% to xm to state that it's a Cauchy sequence. 00:14:10.220 --> 00:14:12.765 align:middle line:90% So what can we do? 00:14:12.765 --> 00:14:13.265 align:middle line:90% Anyone? 00:14:13.265 --> 00:14:18.516 align:middle line:90% 00:14:18.516 --> 00:14:22.098 align:middle line:84% Want y'all to think about it because we want to bound this 00:14:22.098 --> 00:14:23.765 align:middle line:84% and show that this is less than epsilon. 00:14:23.765 --> 00:14:27.560 align:middle line:90% 00:14:27.560 --> 00:14:28.810 align:middle line:90% AUDIENCE: Triangle inequality. 00:14:28.810 --> 00:14:29.727 align:middle line:90% PAIGE BRIGHT: Exactly. 00:14:29.727 --> 00:14:30.740 align:middle line:90% The triangle inequality. 00:14:30.740 --> 00:14:32.823 align:middle line:84% This is less than or equal to the distance from xn 00:14:32.823 --> 00:14:40.720 align:middle line:84% to x plus the distance from x to xm. 00:14:40.720 --> 00:14:42.925 align:middle line:84% And we can make both of these less than epsilon. 00:14:42.925 --> 00:14:45.397 align:middle line:90% 00:14:45.397 --> 00:14:47.980 align:middle line:84% Now, if I was really careful, I could have made these epsilons 00:14:47.980 --> 00:14:48.970 align:middle line:90% over 2. 00:14:48.970 --> 00:14:50.887 align:middle line:84% And then this would have been exactly epsilon. 00:14:50.887 --> 00:14:53.428 align:middle line:84% Most of the time, if you show it's less than a constant times 00:14:53.428 --> 00:14:54.310 align:middle line:90% epsilon, you're good. 00:14:54.310 --> 00:14:55.893 align:middle line:84% You would just have to relabel things. 00:14:55.893 --> 00:14:57.670 align:middle line:90% So all good things here. 00:14:57.670 --> 00:14:58.510 align:middle line:90% But yeah, exactly. 00:14:58.510 --> 00:15:01.190 align:middle line:84% You apply the triangle inequality. 00:15:01.190 --> 00:15:01.690 align:middle line:90% Oh. 00:15:01.690 --> 00:15:03.400 align:middle line:84% And then this is precisely what we want 00:15:03.400 --> 00:15:04.600 align:middle line:90% for Cauchy sequence, right? 00:15:04.600 --> 00:15:09.150 align:middle line:84% This implies that it is, in fact, a Cauchy sequence. 00:15:09.150 --> 00:15:11.670 align:middle line:90% So we're done. 00:15:11.670 --> 00:15:14.780 align:middle line:84% Now, the real question is, if Cauchy sequences are convergent 00:15:14.780 --> 00:15:19.100 align:middle line:84% sequences-- and as we know from the real numbers, 00:15:19.100 --> 00:15:21.470 align:middle line:84% from the real numbers, we know that there it's Cauchy 00:15:21.470 --> 00:15:23.067 align:middle line:90% complete, but here we don't. 00:15:23.067 --> 00:15:24.650 align:middle line:84% And what do I mean by Cauchy complete, 00:15:24.650 --> 00:15:26.570 align:middle line:90% which I've stated a few times? 00:15:26.570 --> 00:15:39.680 align:middle line:84% Definition-- a space is Cauchy complete if and only if-- i.e. 00:15:39.680 --> 00:15:48.010 align:middle line:84% the definition-- the Cauchy sequences are convergent. 00:15:48.010 --> 00:15:53.313 align:middle line:90% 00:15:53.313 --> 00:15:55.730 align:middle line:84% So this is the definition of what Cauchy complete actually 00:15:55.730 --> 00:15:56.420 align:middle line:90% means. 00:15:56.420 --> 00:15:58.460 align:middle line:84% And this is a very powerful fact, right? 00:15:58.460 --> 00:16:01.970 align:middle line:84% We use Cauchy completeness everywhere in real analysis 00:16:01.970 --> 00:16:03.620 align:middle line:84% to state that limit points exist, 00:16:03.620 --> 00:16:05.600 align:middle line:84% to state that differentiability was nice. 00:16:05.600 --> 00:16:08.450 align:middle line:84% Things like that-- we needed Cauchy completeness there. 00:16:08.450 --> 00:16:11.060 align:middle line:84% Now, not every space is Cauchy complete, unfortunately, 00:16:11.060 --> 00:16:14.510 align:middle line:84% but the ones that we really want them to be are, in fact. 00:16:14.510 --> 00:16:19.070 align:middle line:84% So proposition-- this is on your homework-- 00:16:19.070 --> 00:16:27.260 align:middle line:84% we have that yeah, the set of continuous functions on 0 to 1 00:16:27.260 --> 00:16:28.160 align:middle line:90% is Cauchy complete. 00:16:28.160 --> 00:16:34.360 align:middle line:90% 00:16:34.360 --> 00:16:36.970 align:middle line:84% This is a statement on your homework 00:16:36.970 --> 00:16:39.110 align:middle line:84% that I've broken up into individual steps. 00:16:39.110 --> 00:16:41.950 align:middle line:84% So this should be a little bit nicer. 00:16:41.950 --> 00:16:45.775 align:middle line:84% But yeah, a very useful proposition to show. 00:16:45.775 --> 00:16:47.170 align:middle line:90% OK. 00:16:47.170 --> 00:16:51.000 align:middle line:84% Now, let me go back to what I meant to do slightly earlier. 00:16:51.000 --> 00:16:51.990 align:middle line:90% OK. 00:16:51.990 --> 00:16:55.560 align:middle line:84% We have a few more things to say about convergent sequences. 00:16:55.560 --> 00:16:57.570 align:middle line:84% And to do so, I'm going to define what it means 00:16:57.570 --> 00:16:59.010 align:middle line:90% for a set to be bounded, right? 00:16:59.010 --> 00:17:00.660 align:middle line:84% When we are looking at real analysis, 00:17:00.660 --> 00:17:03.090 align:middle line:84% one of the most useful theorems we had in our tool belt 00:17:03.090 --> 00:17:05.085 align:middle line:84% was the Bolzano-Weierstrass theorem. 00:17:05.085 --> 00:17:06.960 align:middle line:84% Now, we're not going to have an analog of it, 00:17:06.960 --> 00:17:09.030 align:middle line:84% but we still have some statements akin to it 00:17:09.030 --> 00:17:10.499 align:middle line:90% that we want to show. 00:17:10.499 --> 00:17:12.180 align:middle line:84% And this will be especially important 00:17:12.180 --> 00:17:14.790 align:middle line:90% as we build up to compact sets. 00:17:14.790 --> 00:17:27.060 align:middle line:84% So definition-- a sequence xn is bounded by, let's say, 00:17:27.060 --> 00:17:33.850 align:middle line:84% b bigger than 0 if, for all n in the natural numbers-- 00:17:33.850 --> 00:17:37.990 align:middle line:90% 00:17:37.990 --> 00:17:38.490 align:middle line:90% oh, sorry. 00:17:38.490 --> 00:17:44.970 align:middle line:84% I should say if there exists a point p in your metrics 00:17:44.970 --> 00:17:50.250 align:middle line:84% base such that for all n in the natural numbers 00:17:50.250 --> 00:17:54.360 align:middle line:84% the distance from xn to p is less than B-- 00:17:54.360 --> 00:17:55.690 align:middle line:90% which is what we would expect. 00:17:55.690 --> 00:17:59.880 align:middle line:84% We want the sequence itself to be-- 00:17:59.880 --> 00:18:03.690 align:middle line:84% the distance between the point and your sequence 00:18:03.690 --> 00:18:04.650 align:middle line:90% to be bounded. 00:18:04.650 --> 00:18:07.380 align:middle line:84% That's what it means for a sequence to be bounded. 00:18:07.380 --> 00:18:10.500 align:middle line:84% And if you want to picture this, I always 00:18:10.500 --> 00:18:12.960 align:middle line:84% like to draw little pictures in my notes about this. 00:18:12.960 --> 00:18:16.230 align:middle line:84% If this is our metric space x, here's our point p. 00:18:16.230 --> 00:18:18.480 align:middle line:84% And what this is stating is that there 00:18:18.480 --> 00:18:26.640 align:middle line:84% exists a radius large enough of radius B such that it 00:18:26.640 --> 00:18:29.210 align:middle line:90% completely contains x. 00:18:29.210 --> 00:18:31.760 align:middle line:84% Now, granted, what does it mean to have stuff outside of x? 00:18:31.760 --> 00:18:32.510 align:middle line:90% Nothing, really. 00:18:32.510 --> 00:18:36.740 align:middle line:84% So really, we just mean the region in the middle 00:18:36.740 --> 00:18:39.860 align:middle line:84% is contained in the ball of radius B. 00:18:39.860 --> 00:18:42.470 align:middle line:84% But that's a nice picture, if you prefer. 00:18:42.470 --> 00:18:47.610 align:middle line:84% And not only can we say this about sequences in general, 00:18:47.610 --> 00:18:50.405 align:middle line:84% we can also define what this means for a set. 00:18:50.405 --> 00:19:09.740 align:middle line:84% A set A in x is bounded by B if, for all p in x-- 00:19:09.740 --> 00:19:10.240 align:middle line:90% oh, sorry. 00:19:10.240 --> 00:19:15.040 align:middle line:84% If there exists a p in x such that for all A 00:19:15.040 --> 00:19:20.650 align:middle line:84% in A the distance from A to p is less 00:19:20.650 --> 00:19:25.485 align:middle line:84% than B. Does the set theory notation that I'm 00:19:25.485 --> 00:19:26.610 align:middle line:90% using make sense, everyone? 00:19:26.610 --> 00:19:28.740 align:middle line:90% There exists, and for all? 00:19:28.740 --> 00:19:29.560 align:middle line:90% Cool. 00:19:29.560 --> 00:19:31.560 align:middle line:84% So this is what it means for a sequence in a set 00:19:31.560 --> 00:19:32.247 align:middle line:90% to be bounded. 00:19:32.247 --> 00:19:33.330 align:middle line:90% Why would I bring this up? 00:19:33.330 --> 00:19:36.270 align:middle line:84% Because convergent sequences are bounded sequences. 00:19:36.270 --> 00:19:38.130 align:middle line:90% We have this from real analysis. 00:19:38.130 --> 00:19:41.430 align:middle line:84% But I'm going to prove it now in metric spaces. 00:19:41.430 --> 00:19:48.690 align:middle line:84% So proposition-- or I'll just write it 00:19:48.690 --> 00:19:50.670 align:middle line:84% out how I've been writing out the other ones. 00:19:50.670 --> 00:19:58.185 align:middle line:84% If xn converges to x, then xn is the bounded sequence. 00:19:58.185 --> 00:20:02.960 align:middle line:90% 00:20:02.960 --> 00:20:04.472 align:middle line:90% OK. 00:20:04.472 --> 00:20:06.680 align:middle line:84% Let's start off with the proof, which will start off, 00:20:06.680 --> 00:20:08.720 align:middle line:84% as all of our statements have so far, 00:20:08.720 --> 00:20:12.898 align:middle line:84% with the statement of what convergence means. 00:20:12.898 --> 00:20:14.690 align:middle line:84% In the statement of what convergence means, 00:20:14.690 --> 00:20:15.680 align:middle line:84% we're going to choose the epsilon 00:20:15.680 --> 00:20:17.138 align:middle line:84% that I want so that I'm not dealing 00:20:17.138 --> 00:20:18.600 align:middle line:90% with all the possible epsilons. 00:20:18.600 --> 00:20:25.310 align:middle line:84% So I'm going to note for epsilon equal to 1 bigger than 0, 00:20:25.310 --> 00:20:28.550 align:middle line:84% we know that there exists an n in the natural numbers 00:20:28.550 --> 00:20:36.800 align:middle line:84% such that for all n bigger than or equal to n, 00:20:36.800 --> 00:20:42.570 align:middle line:84% the distance from xn to x is less than epsilon, 00:20:42.570 --> 00:20:45.060 align:middle line:90% which again, I'm assuming is 1. 00:20:45.060 --> 00:20:47.620 align:middle line:84% Now, does this complete our statement? 00:20:47.620 --> 00:20:49.170 align:middle line:90% The answer is no. 00:20:49.170 --> 00:20:52.650 align:middle line:84% Here we only have that this is true for all n bigger than 00:20:52.650 --> 00:20:55.350 align:middle line:90% or equal to capital N, right? 00:20:55.350 --> 00:20:58.650 align:middle line:84% But what we want for our statement is 00:20:58.650 --> 00:21:01.110 align:middle line:84% to show that for every single natural number, 00:21:01.110 --> 00:21:02.580 align:middle line:90% this sequence is bounded. 00:21:02.580 --> 00:21:06.270 align:middle line:84% But this isn't an issue because what we can note 00:21:06.270 --> 00:21:09.330 align:middle line:84% is that there are finitely many terms less than n 00:21:09.330 --> 00:21:11.700 align:middle line:90% in the natural numbers, right? 00:21:11.700 --> 00:21:17.610 align:middle line:84% So what we can do is that B be the maximum 00:21:17.610 --> 00:21:22.080 align:middle line:84% of the finitely many terms of the distance 00:21:22.080 --> 00:21:34.830 align:middle line:84% from xi to x, or, let's say, 1 for i between 1 00:21:34.830 --> 00:21:38.010 align:middle line:84% and capital N. So what I'm doing here 00:21:38.010 --> 00:21:41.460 align:middle line:84% is I'm noting that here I have my convergent sequence. 00:21:41.460 --> 00:21:44.500 align:middle line:90% 00:21:44.500 --> 00:21:46.450 align:middle line:90% I know that most of them-- 00:21:46.450 --> 00:21:48.520 align:middle line:84% in fact, infinitely many of them-- 00:21:48.520 --> 00:21:54.320 align:middle line:84% are contained in a ball of radius 1. 00:21:54.320 --> 00:21:58.962 align:middle line:84% And I only have finitely many terms outside of that ball. 00:21:58.962 --> 00:22:00.920 align:middle line:84% So what I'm doing is I'm saying, OK, well, it's 00:22:00.920 --> 00:22:05.690 align:middle line:84% either in this ball, or in the next one, or the next one, 00:22:05.690 --> 00:22:07.520 align:middle line:90% or so on and so forth. 00:22:07.520 --> 00:22:10.700 align:middle line:84% This is what the statement is that we wanted to show. 00:22:10.700 --> 00:22:13.880 align:middle line:84% And we know that this is, in fact, finite, 00:22:13.880 --> 00:22:16.050 align:middle line:84% as there is only finitely many terms. 00:22:16.050 --> 00:22:19.202 align:middle line:84% So this is the B that it's bounded by. 00:22:19.202 --> 00:22:21.410 align:middle line:84% I can write that out some more if y'all would prefer, 00:22:21.410 --> 00:22:24.510 align:middle line:84% but this is essentially the statement. 00:22:24.510 --> 00:22:26.480 align:middle line:90% Cool. 00:22:26.480 --> 00:22:29.803 align:middle line:90% So I think that's mostly-- 00:22:29.803 --> 00:22:31.220 align:middle line:84% oh, there's one more thing we want 00:22:31.220 --> 00:22:34.550 align:middle line:84% to say about convergent sequences, which 00:22:34.550 --> 00:22:37.010 align:middle line:84% is that their subsequences act nicely, 00:22:37.010 --> 00:22:40.290 align:middle line:84% which will be essentially how we want it to be. 00:22:40.290 --> 00:22:52.096 align:middle line:84% So proposition-- let xn converge to x and xnk 00:22:52.096 --> 00:22:58.320 align:middle line:90% be a subsequence of xn. 00:22:58.320 --> 00:23:01.370 align:middle line:90% 00:23:01.370 --> 00:23:08.420 align:middle line:84% Then my claim is that xnk is convergent. 00:23:08.420 --> 00:23:12.890 align:middle line:90% 00:23:12.890 --> 00:23:14.738 align:middle line:84% And in fact, to show that it's convergent, 00:23:14.738 --> 00:23:16.280 align:middle line:84% we're going to show that it converges 00:23:16.280 --> 00:23:19.820 align:middle line:84% to x, which is what we should expect of a subsequence. 00:23:19.820 --> 00:23:22.010 align:middle line:84% Does everyone here know what a subsequence is? 00:23:22.010 --> 00:23:25.420 align:middle line:90% Happy to redefine. 00:23:25.420 --> 00:23:27.610 align:middle line:90% Cool. 00:23:27.610 --> 00:23:29.660 align:middle line:90% So let's go with this statement. 00:23:29.660 --> 00:23:33.393 align:middle line:84% So proof-- notice that my subsequence here is arbitrary. 00:23:33.393 --> 00:23:34.810 align:middle line:84% I'm not going to rewrite that out, 00:23:34.810 --> 00:23:37.630 align:middle line:84% but we could choose any subsequence of our sequence. 00:23:37.630 --> 00:23:42.970 align:middle line:84% And what we're interested in is we want to show for all epsilon 00:23:42.970 --> 00:23:47.170 align:middle line:84% bigger than 0, there exists an n in the natural numbers 00:23:47.170 --> 00:23:54.200 align:middle line:84% such that for all nk now bigger than or equal to n, 00:23:54.200 --> 00:23:58.880 align:middle line:84% the distance from xnk to x is less than epsilon. 00:23:58.880 --> 00:24:01.480 align:middle line:90% 00:24:01.480 --> 00:24:05.490 align:middle line:84% And now, again, any guesses as to what we should do? 00:24:05.490 --> 00:24:06.330 align:middle line:90% Triangle inequality. 00:24:06.330 --> 00:24:07.290 align:middle line:90% Precisely. 00:24:07.290 --> 00:24:07.860 align:middle line:90% Yeah. 00:24:07.860 --> 00:24:12.240 align:middle line:84% So we're considering the distance from xnk to x. 00:24:12.240 --> 00:24:15.000 align:middle line:84% What we can do is apply the fact that we know that convergent 00:24:15.000 --> 00:24:17.640 align:middle line:90% sequences are Cauchy. 00:24:17.640 --> 00:24:21.240 align:middle line:84% So we can write this as less than 00:24:21.240 --> 00:24:24.210 align:middle line:84% or equal to the distance from xnk, 00:24:24.210 --> 00:24:31.830 align:middle line:84% let's say, to xm for m bigger than or equal to n 00:24:31.830 --> 00:24:37.020 align:middle line:90% plus the distance from xm to x. 00:24:37.020 --> 00:24:39.630 align:middle line:84% And now, I can choose a larger natural number, if necessary, 00:24:39.630 --> 00:24:42.420 align:middle line:84% such that this term gets less than epsilon over 2. 00:24:42.420 --> 00:24:45.720 align:middle line:84% And this term, because it's Cauchy, is less than epsilon 00:24:45.720 --> 00:24:46.920 align:middle line:90% over 2. 00:24:46.920 --> 00:24:49.110 align:middle line:84% So without writing all those steps, 00:24:49.110 --> 00:24:50.295 align:middle line:90% this is less than epsilon. 00:24:50.295 --> 00:24:53.260 align:middle line:90% 00:24:53.260 --> 00:24:55.620 align:middle line:84% So this shows that the distance from xnk to x 00:24:55.620 --> 00:24:57.100 align:middle line:90% gets arbitrarily small. 00:24:57.100 --> 00:24:58.410 align:middle line:90% So therefore, it's convergent. 00:24:58.410 --> 00:25:01.300 align:middle line:90% 00:25:01.300 --> 00:25:03.220 align:middle line:90% Yeah. 00:25:03.220 --> 00:25:07.150 align:middle line:90% This is fairly important. 00:25:07.150 --> 00:25:08.740 align:middle line:84% And it highlights one of the many ways 00:25:08.740 --> 00:25:11.530 align:middle line:84% in which Cauchy sequences are deeply important, as we'll 00:25:11.530 --> 00:25:14.560 align:middle line:84% talk about in the fifth lecture in the specific module 00:25:14.560 --> 00:25:16.540 align:middle line:90% of this class. 00:25:16.540 --> 00:25:18.495 align:middle line:84% Having things be Cauchy complete-- really, 00:25:18.495 --> 00:25:19.120 align:middle line:90% really helpful. 00:25:19.120 --> 00:25:23.400 align:middle line:90% 00:25:23.400 --> 00:25:24.930 align:middle line:90% OK. 00:25:24.930 --> 00:25:25.950 align:middle line:90% Oh, one more note. 00:25:25.950 --> 00:25:29.070 align:middle line:84% This is something that I vaguely noted on Tuesday, 00:25:29.070 --> 00:25:33.240 align:middle line:84% but these are all of the major theorems and propositions 00:25:33.240 --> 00:25:35.610 align:middle line:84% that we're going to need for convergent sequences. 00:25:35.610 --> 00:25:37.830 align:middle line:84% You might be wondering why there aren't more. 00:25:37.830 --> 00:25:39.760 align:middle line:84% We had a ton more in real analysis. 00:25:39.760 --> 00:25:42.260 align:middle line:84% In fact, it took a month of our time. 00:25:42.260 --> 00:25:44.940 align:middle line:84% The reason we don't have more is because the real numbers 00:25:44.940 --> 00:25:46.238 align:middle line:90% are a vector space. 00:25:46.238 --> 00:25:48.280 align:middle line:84% I can add two real numbers and get a real number. 00:25:48.280 --> 00:25:50.130 align:middle line:84% I can multiply them and get a real number. 00:25:50.130 --> 00:25:53.290 align:middle line:84% Everything there was nicer, and we had addition defined. 00:25:53.290 --> 00:25:56.220 align:middle line:84% So one thing we could state in real analysis 00:25:56.220 --> 00:25:59.430 align:middle line:84% is that for instance, the sum of two convergent sequences 00:25:59.430 --> 00:26:00.480 align:middle line:90% is convergent. 00:26:00.480 --> 00:26:02.295 align:middle line:84% Here that doesn't make sense, necessarily. 00:26:02.295 --> 00:26:05.520 align:middle line:84% You can't always add two points in a metric space 00:26:05.520 --> 00:26:07.500 align:middle line:84% and get a point in your metric space, 00:26:07.500 --> 00:26:10.590 align:middle line:84% let alone have addition be well defined. 00:26:10.590 --> 00:26:13.290 align:middle line:84% Similarly, we have the squeeze theorem in real analysis. 00:26:13.290 --> 00:26:16.620 align:middle line:84% We have a version of it based off of distances, 00:26:16.620 --> 00:26:20.100 align:middle line:84% but we don't have a squeeze theorem for points 00:26:20.100 --> 00:26:22.630 align:middle line:84% because we don't necessarily have an ordering on our set, 00:26:22.630 --> 00:26:23.130 align:middle line:90% right? 00:26:23.130 --> 00:26:24.810 align:middle line:84% You can picture the complex numbers. 00:26:24.810 --> 00:26:28.482 align:middle line:84% The complex numbers don't have an ordering on them. 00:26:28.482 --> 00:26:30.690 align:middle line:84% You don't have a sense in which one complex number is 00:26:30.690 --> 00:26:32.880 align:middle line:90% bigger than the other. 00:26:32.880 --> 00:26:34.797 align:middle line:84% Yeah, I'm just going to leave it there. 00:26:34.797 --> 00:26:36.630 align:middle line:84% If you've done complex analysis, then you'll 00:26:36.630 --> 00:26:39.210 align:middle line:90% know what I'm talking about. 00:26:39.210 --> 00:26:40.410 align:middle line:90% But yeah. 00:26:40.410 --> 00:26:43.180 align:middle line:84% So that's why we don't have more theorems as we used to. 00:26:43.180 --> 00:26:45.840 align:middle line:84% So now, we're going to move on to the next major theorem 00:26:45.840 --> 00:26:48.450 align:middle line:84% that I've talked about, or the next major definition that I've 00:26:48.450 --> 00:26:48.930 align:middle line:90% talked about. 00:26:48.930 --> 00:26:50.610 align:middle line:84% We've talked about convergent sequences. 00:26:50.610 --> 00:26:53.010 align:middle line:84% We've talked a little bit about Cauchy sequences. 00:26:53.010 --> 00:26:55.740 align:middle line:84% Cauchy sequences are helpful, but in terms of a metric space, 00:26:55.740 --> 00:26:59.280 align:middle line:84% we've mostly stated what we need for the moment being. 00:26:59.280 --> 00:27:01.023 align:middle line:84% Now, we're going to move on to open sets. 00:27:01.023 --> 00:27:02.940 align:middle line:84% So I'm going to recall what this definition is 00:27:02.940 --> 00:27:05.670 align:middle line:84% because it's the one that can be a little bit the weirdest. 00:27:05.670 --> 00:27:19.440 align:middle line:84% So recall-- a set A contained in your metric space x is open 00:27:19.440 --> 00:27:24.720 align:middle line:84% if, for every single point in A, there exists an epsilon 00:27:24.720 --> 00:27:28.890 align:middle line:84% bigger than 0 such that the ball of radius epsilon around A 00:27:28.890 --> 00:27:31.530 align:middle line:84% is completely contained in your set. 00:27:31.530 --> 00:27:36.120 align:middle line:84% And this is the set of y in your metric space such 00:27:36.120 --> 00:27:41.707 align:middle line:84% that the distance from y to A is less than epsilon. 00:27:41.707 --> 00:27:43.290 align:middle line:84% Just to bring it up because sometimes, 00:27:43.290 --> 00:27:45.720 align:middle line:84% the notation comes up, this is also 00:27:45.720 --> 00:27:50.310 align:middle line:84% sometimes denoted the ball around A of radius epsilon. 00:27:50.310 --> 00:27:54.210 align:middle line:84% So use whichever notation you prefer. 00:27:54.210 --> 00:27:56.320 align:middle line:84% Here-- I can lift it a little bit up. 00:27:56.320 --> 00:27:59.100 align:middle line:84% So use whichever notation you prefer. 00:27:59.100 --> 00:28:02.130 align:middle line:84% I prefer this one just because it means 00:28:02.130 --> 00:28:05.880 align:middle line:90% less commas, but whichever. 00:28:05.880 --> 00:28:07.560 align:middle line:90% OK. 00:28:07.560 --> 00:28:11.760 align:middle line:84% So open sets are going to have a huge connection 00:28:11.760 --> 00:28:14.430 align:middle line:84% between topology, and continuity, 00:28:14.430 --> 00:28:17.823 align:middle line:84% and other definitions that are important. 00:28:17.823 --> 00:28:19.740 align:middle line:84% Now, the first thing that we're going to prove 00:28:19.740 --> 00:28:22.480 align:middle line:84% is topological properties of open sets. 00:28:22.480 --> 00:28:25.180 align:middle line:84% So because y'all haven't taken topology, this is, in fact, 00:28:25.180 --> 00:28:26.200 align:middle line:90% really, really helpful. 00:28:26.200 --> 00:28:29.310 align:middle line:84% We're going to show three major properties of open sets 00:28:29.310 --> 00:28:31.090 align:middle line:90% in metric spaces. 00:28:31.090 --> 00:28:37.660 align:middle line:84% So a theorem-- and you can call this topological properties, 00:28:37.660 --> 00:28:40.120 align:middle line:90% if you want. 00:28:40.120 --> 00:28:41.590 align:middle line:90% Or I'll write it out because-- 00:28:41.590 --> 00:28:49.955 align:middle line:84% topological properties of open sets. 00:28:49.955 --> 00:28:54.495 align:middle line:90% 00:28:54.495 --> 00:28:54.995 align:middle line:90% OK. 00:28:54.995 --> 00:28:58.130 align:middle line:84% Firstly, we have that the empty set 00:28:58.130 --> 00:29:01.685 align:middle line:84% and the entire metric space itself are open. 00:29:01.685 --> 00:29:04.950 align:middle line:90% 00:29:04.950 --> 00:29:19.700 align:middle line:84% Two, given Ai are open sets, then the union of all of these 00:29:19.700 --> 00:29:23.030 align:middle line:84% sets from i equals, let's say, 1 to infinity-- 00:29:23.030 --> 00:29:25.090 align:middle line:90% or I'll just use 0, why not-- 00:29:25.090 --> 00:29:26.210 align:middle line:90% is open. 00:29:26.210 --> 00:29:31.610 align:middle line:84% So in other words, the arbitrary union of open sets is open. 00:29:31.610 --> 00:29:45.900 align:middle line:84% And finally, the finite intersections of open sets 00:29:45.900 --> 00:29:46.400 align:middle line:90% are open. 00:29:46.400 --> 00:29:51.260 align:middle line:90% 00:29:51.260 --> 00:29:54.260 align:middle line:84% In other words, if I intersect from i 00:29:54.260 --> 00:30:01.270 align:middle line:84% equals 0 to capital M of Ai, then this is open. 00:30:01.270 --> 00:30:04.030 align:middle line:84% The fact that we can't intersect infinitely many of them 00:30:04.030 --> 00:30:06.650 align:middle line:84% will become apparent in the proof. 00:30:06.650 --> 00:30:08.680 align:middle line:84% So in the proof of these three properties, 00:30:08.680 --> 00:30:10.180 align:middle line:84% it will become apparent why we can't 00:30:10.180 --> 00:30:12.520 align:middle line:90% do infinitely many of them. 00:30:12.520 --> 00:30:13.330 align:middle line:90% But let's see this. 00:30:13.330 --> 00:30:18.670 align:middle line:90% 00:30:18.670 --> 00:30:21.970 align:middle line:90% One, so consider the empty set. 00:30:21.970 --> 00:30:24.040 align:middle line:84% How do we know that the empty set is open? 00:30:24.040 --> 00:30:26.620 align:middle line:90% Well, it's vacuously true. 00:30:26.620 --> 00:30:28.990 align:middle line:84% It's true all the time because for every single point 00:30:28.990 --> 00:30:31.430 align:middle line:84% in the empty set, there are no points in the empty set. 00:30:31.430 --> 00:30:33.490 align:middle line:84% So as soon as that statement's false, 00:30:33.490 --> 00:30:36.370 align:middle line:84% we have that the empty set is open. 00:30:36.370 --> 00:30:38.620 align:middle line:90% So this is open vacuously. 00:30:38.620 --> 00:30:41.500 align:middle line:90% 00:30:41.500 --> 00:30:43.840 align:middle line:90% Two, let's consider the set x. 00:30:43.840 --> 00:30:47.740 align:middle line:90% Well, for all epsilon-- 00:30:47.740 --> 00:30:50.320 align:middle line:84% or sorry, for every single point x 00:30:50.320 --> 00:30:54.250 align:middle line:84% in x, does there exist a ball around x in x? 00:30:54.250 --> 00:30:59.500 align:middle line:84% Yes, there is because just pick your favorite epsilon bigger 00:30:59.500 --> 00:31:00.580 align:middle line:90% than 0. 00:31:00.580 --> 00:31:03.550 align:middle line:84% And then the ball of radius epsilon around that point 00:31:03.550 --> 00:31:07.570 align:middle line:84% x is, by definition, a subset of your metric space, right? 00:31:07.570 --> 00:31:10.870 align:middle line:84% Because the definition of a ball of radius epsilon 00:31:10.870 --> 00:31:12.117 align:middle line:90% is only the point in x. 00:31:12.117 --> 00:31:14.200 align:middle line:84% So we're not going to end up with some weird thing 00:31:14.200 --> 00:31:16.660 align:middle line:90% outside of x. 00:31:16.660 --> 00:31:21.040 align:middle line:84% So this shows that x itself is open. 00:31:21.040 --> 00:31:24.100 align:middle line:84% Two, we want to show that the union of open sets is open. 00:31:24.100 --> 00:31:25.060 align:middle line:90% How do we do that? 00:31:25.060 --> 00:31:28.250 align:middle line:90% 00:31:28.250 --> 00:31:33.210 align:middle line:90% i equals 0 to infinity is open. 00:31:33.210 --> 00:31:36.330 align:middle line:90% This is what we want to show. 00:31:36.330 --> 00:31:38.533 align:middle line:84% Well, as with the definition, we're 00:31:38.533 --> 00:31:40.200 align:middle line:84% just going to pick a point in the union. 00:31:40.200 --> 00:31:45.420 align:middle line:84% So pick arbitrary x in the union-- 00:31:45.420 --> 00:31:49.038 align:middle line:90% equals 0 to infinity of Ai. 00:31:49.038 --> 00:31:50.580 align:middle line:84% And we want to show that there exists 00:31:50.580 --> 00:31:54.900 align:middle line:84% a ball of radius epsilon around x that's contained in the set. 00:31:54.900 --> 00:31:56.730 align:middle line:84% Well, because it's in the union, there 00:31:56.730 --> 00:31:59.970 align:middle line:84% has to exist at least 1 Ai that contains x. 00:31:59.970 --> 00:32:05.880 align:middle line:84% So therefore, there exists an Aj in your set of open sets 00:32:05.880 --> 00:32:11.850 align:middle line:90% Ai such that x is in Aj. 00:32:11.850 --> 00:32:15.420 align:middle line:84% And then we apply the fact that Aj itself is open. 00:32:15.420 --> 00:32:19.710 align:middle line:84% So therefore, there exists an epsilon bigger than 0 such 00:32:19.710 --> 00:32:28.680 align:middle line:84% that the ball of radius epsilon around x is contained in Aj, 00:32:28.680 --> 00:32:30.780 align:middle line:84% but this is a subset of the union. 00:32:30.780 --> 00:32:34.175 align:middle line:90% 00:32:34.175 --> 00:32:36.050 align:middle line:84% So this shows that the ball of radius epsilon 00:32:36.050 --> 00:32:38.750 align:middle line:84% is contained in the infinite union, or potentially 00:32:38.750 --> 00:32:39.860 align:middle line:90% infinite union. 00:32:39.860 --> 00:32:42.800 align:middle line:84% This also works if it's finite, notice. 00:32:42.800 --> 00:32:46.060 align:middle line:90% But that's just a subcase. 00:32:46.060 --> 00:32:47.380 align:middle line:90% OK. 00:32:47.380 --> 00:32:50.110 align:middle line:90% And can I squeeze it in here? 00:32:50.110 --> 00:32:50.740 align:middle line:90% No, I cannot. 00:32:50.740 --> 00:32:54.500 align:middle line:90% So I will move over here. 00:32:54.500 --> 00:32:55.810 align:middle line:90% Any questions so far? 00:32:55.810 --> 00:32:57.250 align:middle line:90% Actually, let me move this. 00:32:57.250 --> 00:32:58.210 align:middle line:90% Any questions? 00:32:58.210 --> 00:33:01.760 align:middle line:90% 00:33:01.760 --> 00:33:06.130 align:middle line:84% Open sets are weird in that when we 00:33:06.130 --> 00:33:07.940 align:middle line:84% learn about them in real analysis, 00:33:07.940 --> 00:33:11.380 align:middle line:84% I feel like they're pretty unintuitively important. 00:33:11.380 --> 00:33:16.740 align:middle line:84% Everyone says that they're important, but I never got why. 00:33:16.740 --> 00:33:21.690 align:middle line:84% And I'll explain why they are just in a moment. 00:33:21.690 --> 00:33:24.810 align:middle line:84% But yeah, open sets are deeply important in real analysis 00:33:24.810 --> 00:33:26.610 align:middle line:90% and topology. 00:33:26.610 --> 00:33:29.040 align:middle line:84% And we're going to prove some more statements 00:33:29.040 --> 00:33:31.890 align:middle line:84% besides these topological properties that 00:33:31.890 --> 00:33:37.020 align:middle line:84% will be important and will relate to the major definitions 00:33:37.020 --> 00:33:37.840 align:middle line:90% in our toolbox. 00:33:37.840 --> 00:33:41.724 align:middle line:90% 00:33:41.724 --> 00:33:43.660 align:middle line:90% OK. 00:33:43.660 --> 00:33:48.460 align:middle line:84% So now, we're interested in the third case, 00:33:48.460 --> 00:33:52.590 align:middle line:84% the intersection from i equals 0 to m of the Ai. 00:33:52.590 --> 00:33:56.140 align:middle line:84% Well, we're going to do exactly as we did before. 00:33:56.140 --> 00:34:00.690 align:middle line:84% If I have some x in the intersection of the Ai from i 00:34:00.690 --> 00:34:04.380 align:middle line:84% equals 0 to m, then what do I know? 00:34:04.380 --> 00:34:07.140 align:middle line:84% Before, we knew that it was in any of the Ai's. 00:34:07.140 --> 00:34:09.389 align:middle line:84% But here we know even more information. 00:34:09.389 --> 00:34:17.389 align:middle line:84% Here we know that x is in Ai for every single i. 00:34:17.389 --> 00:34:18.600 align:middle line:90% So what can we do? 00:34:18.600 --> 00:34:23.449 align:middle line:84% Well, we know that therefore, because each Ai is open, 00:34:23.449 --> 00:34:27.920 align:middle line:84% there exists an epsilon i bigger than 0 such 00:34:27.920 --> 00:34:35.190 align:middle line:84% that the ball of radius epsilon i around x is contained in Ai. 00:34:35.190 --> 00:34:38.010 align:middle line:84% The issue is that the entirety of Ai 00:34:38.010 --> 00:34:39.820 align:middle line:84% might not be in the intersection. 00:34:39.820 --> 00:34:41.040 align:middle line:90% So what can we do? 00:34:41.040 --> 00:34:43.350 align:middle line:90% We have this for every single i. 00:34:43.350 --> 00:34:44.880 align:middle line:84% And we have finitely many of them. 00:34:44.880 --> 00:34:53.889 align:middle line:84% So choose epsilon to be equal to the minimum of the epsilon 00:34:53.889 --> 00:35:00.600 align:middle line:90% i's from i equals 0 or 0 to M-- 00:35:00.600 --> 00:35:04.380 align:middle line:84% capital M. This is bigger than 0, right? 00:35:04.380 --> 00:35:06.820 align:middle line:84% Because we're only choosing finite many. 00:35:06.820 --> 00:35:10.260 align:middle line:84% This is where we're using the fact that there's 00:35:10.260 --> 00:35:11.640 align:middle line:90% the finite intersection. 00:35:11.640 --> 00:35:14.130 align:middle line:84% If you had an infinite intersection, 00:35:14.130 --> 00:35:18.420 align:middle line:84% you can picture taking more and more, smaller and smaller 00:35:18.420 --> 00:35:22.470 align:middle line:84% epsilon i's such that this infimum is 0. 00:35:22.470 --> 00:35:24.810 align:middle line:84% And that's an issue, but if it's finitely many, 00:35:24.810 --> 00:35:27.900 align:middle line:90% we can take the minimum of them. 00:35:27.900 --> 00:35:30.270 align:middle line:84% And then what we know is that the ball of radius 00:35:30.270 --> 00:35:35.250 align:middle line:84% epsilon around x is a subset of Ai for all i 00:35:35.250 --> 00:35:38.946 align:middle line:84% because it's contained in the ball of radius epsilon i. 00:35:38.946 --> 00:35:40.950 align:middle line:84% Or I guess I should say that this is contained 00:35:40.950 --> 00:35:46.800 align:middle line:84% in the ball of radius epsilon i of x, which is contained 00:35:46.800 --> 00:35:50.910 align:middle line:84% in Ai for all i, which implies that the ball of radius 00:35:50.910 --> 00:35:57.610 align:middle line:84% epsilon around x is a subset of the finite intersection. 00:35:57.610 --> 00:35:58.780 align:middle line:90% All right? 00:35:58.780 --> 00:36:01.030 align:middle line:84% So why is this called the topological properties 00:36:01.030 --> 00:36:02.260 align:middle line:90% of open sets? 00:36:02.260 --> 00:36:04.102 align:middle line:84% Because if you were to take 18901, 00:36:04.102 --> 00:36:06.310 align:middle line:84% which is a class that I would highly suggest you take 00:36:06.310 --> 00:36:08.268 align:middle line:84% at some point if you're interested in this sort 00:36:08.268 --> 00:36:11.260 align:middle line:84% of analysis, as we're going to see, 00:36:11.260 --> 00:36:13.960 align:middle line:84% open sets allow us to state things about continuity. 00:36:13.960 --> 00:36:16.300 align:middle line:84% You can define continuity in terms of open sets. 00:36:16.300 --> 00:36:17.950 align:middle line:84% You can define convergent sequences 00:36:17.950 --> 00:36:19.390 align:middle line:90% in terms of open sets. 00:36:19.390 --> 00:36:23.110 align:middle line:84% Basically, everything we can redo in terms of open sets. 00:36:23.110 --> 00:36:25.090 align:middle line:90% And that is a topology. 00:36:25.090 --> 00:36:28.570 align:middle line:84% If I have a set of subsets such that these three 00:36:28.570 --> 00:36:31.330 align:middle line:84% properties hold, that's known as a topology. 00:36:31.330 --> 00:36:34.930 align:middle line:84% And in 18901, that's the basic toolbox that you're given. 00:36:34.930 --> 00:36:38.810 align:middle line:84% And then you go from there in terms of redefining everything. 00:36:38.810 --> 00:36:42.790 align:middle line:84% So this is just an example of a more general thing 00:36:42.790 --> 00:36:45.400 align:middle line:90% than metric spaces. 00:36:45.400 --> 00:36:47.170 align:middle line:84% I'm going to talk a little bit about that 00:36:47.170 --> 00:36:49.960 align:middle line:84% in the final lecture, where I talk about where 00:36:49.960 --> 00:36:51.260 align:middle line:90% things go from here. 00:36:51.260 --> 00:36:55.340 align:middle line:84% But for now, this is the basic idea. 00:36:55.340 --> 00:36:55.840 align:middle line:90% All right. 00:36:55.840 --> 00:36:58.960 align:middle line:90% 00:36:58.960 --> 00:37:01.750 align:middle line:84% Now what I'm going to state is some facts about closed sets. 00:37:01.750 --> 00:37:05.590 align:middle line:84% But what is a closed set, which I haven't defined before? 00:37:05.590 --> 00:37:07.160 align:middle line:90% So this is useful to know. 00:37:07.160 --> 00:37:19.040 align:middle line:84% A subset A in x is closed if the complement of A, 00:37:19.040 --> 00:37:25.460 align:middle line:84% which is known as the metric space x minus A, is open. 00:37:25.460 --> 00:37:28.780 align:middle line:90% 00:37:28.780 --> 00:37:34.600 align:middle line:84% Now, if a set is closed, does that imply that it's not open? 00:37:34.600 --> 00:37:35.140 align:middle line:90% Yeah. 00:37:35.140 --> 00:37:36.280 align:middle line:90% The answer is no. 00:37:36.280 --> 00:37:37.893 align:middle line:90% How can we see that? 00:37:37.893 --> 00:37:40.060 align:middle line:84% Well, let's consider the real numbers as an example. 00:37:40.060 --> 00:37:43.810 align:middle line:90% 00:37:43.810 --> 00:37:47.042 align:middle line:84% Notice-- what's the complement of the empty set 00:37:47.042 --> 00:37:47.875 align:middle line:90% in the real numbers? 00:37:47.875 --> 00:37:50.970 align:middle line:90% 00:37:50.970 --> 00:37:53.268 align:middle line:90% Anyone? 00:37:53.268 --> 00:37:55.060 align:middle line:84% It's a good exercise in thinking about what 00:37:55.060 --> 00:37:57.320 align:middle line:90% the complement means. 00:37:57.320 --> 00:37:59.940 align:middle line:84% It's the real numbers minus the empty set. 00:37:59.940 --> 00:38:03.780 align:middle line:84% So it's just going to be all of the real numbers. 00:38:03.780 --> 00:38:06.920 align:middle line:84% But this, as we've stated before in our first topological 00:38:06.920 --> 00:38:07.910 align:middle line:90% property-- 00:38:07.910 --> 00:38:10.880 align:middle line:90% this is open. 00:38:10.880 --> 00:38:16.730 align:middle line:84% So therefore, the empty set is closed by definition. 00:38:16.730 --> 00:38:20.810 align:middle line:84% But we also know, by our topological properties, again, 00:38:20.810 --> 00:38:22.100 align:middle line:90% that the empty set is open. 00:38:22.100 --> 00:38:25.700 align:middle line:90% 00:38:25.700 --> 00:38:27.050 align:middle line:90% What the heck? 00:38:27.050 --> 00:38:29.210 align:middle line:84% Well, this is just one of those things in topology. 00:38:29.210 --> 00:38:30.590 align:middle line:90% It's weird. 00:38:30.590 --> 00:38:33.260 align:middle line:84% Open sets-- you can prove properties of closed sets 00:38:33.260 --> 00:38:36.710 align:middle line:90% instead, and everything's fine. 00:38:36.710 --> 00:38:48.210 align:middle line:84% I'll note-- there's a notion called connectedness, which 00:38:48.210 --> 00:38:50.610 align:middle line:84% I'll briefly define, but ultimately will be 00:38:50.610 --> 00:38:52.660 align:middle line:90% more of a problem set problem. 00:38:52.660 --> 00:38:56.340 align:middle line:84% But if your metric space is connected-- 00:38:56.340 --> 00:38:59.420 align:middle line:90% let me state it that way. 00:38:59.420 --> 00:39:03.920 align:middle line:84% If your metric space x is connected, 00:39:03.920 --> 00:39:10.430 align:middle line:84% which I'll define in a moment, this is true if and only 00:39:10.430 --> 00:39:18.650 align:middle line:84% if the only open and closed sets-- 00:39:18.650 --> 00:39:22.040 align:middle line:84% so sets that are both open and closed-- 00:39:22.040 --> 00:39:26.818 align:middle line:84% are the empty set and the metric space itself. 00:39:26.818 --> 00:39:29.360 align:middle line:84% Now, what does it mean for your metric space to be connected? 00:39:29.360 --> 00:39:31.068 align:middle line:84% We have a nice picture of it in our heads 00:39:31.068 --> 00:39:33.470 align:middle line:84% because connected is a pretty visual thing, 00:39:33.470 --> 00:39:35.600 align:middle line:90% but this is one definition. 00:39:35.600 --> 00:39:38.400 align:middle line:84% Once you prove this, this is a definition. 00:39:38.400 --> 00:39:39.380 align:middle line:90% How do you prove this? 00:39:39.380 --> 00:39:42.020 align:middle line:84% Well, you prove it based off of the definition of disconnected. 00:39:42.020 --> 00:39:50.630 align:middle line:84% So definition-- x is disconnected by definition 00:39:50.630 --> 00:39:58.220 align:middle line:84% if there exist two open sets, U1 and U2, that are disjoint-- 00:39:58.220 --> 00:40:02.150 align:middle line:90% 00:40:02.150 --> 00:40:17.070 align:middle line:84% disjoint-- and nonempty such that the union of them 00:40:17.070 --> 00:40:18.270 align:middle line:90% is exactly x. 00:40:18.270 --> 00:40:22.300 align:middle line:84% This is what it means for your space to be disconnected. 00:40:22.300 --> 00:40:26.310 align:middle line:84% And from here a set is connected if it is not disconnected. 00:40:26.310 --> 00:40:28.290 align:middle line:90% So it's circular reasoning. 00:40:28.290 --> 00:40:29.790 align:middle line:84% And then once you have that, you can 00:40:29.790 --> 00:40:33.687 align:middle line:84% prove this note, which I'll put in the third problem set, which 00:40:33.687 --> 00:40:34.770 align:middle line:90% is an interesting problem. 00:40:34.770 --> 00:40:38.040 align:middle line:90% 00:40:38.040 --> 00:40:40.200 align:middle line:84% But I just want to note-- showing 00:40:40.200 --> 00:40:41.970 align:middle line:84% something is closed does not necessarily 00:40:41.970 --> 00:40:45.060 align:middle line:90% imply that it's open. 00:40:45.060 --> 00:40:47.640 align:middle line:84% I'll give one short example of why this is true. 00:40:47.640 --> 00:40:51.450 align:middle line:84% And then I'll move on to more properties of open sets. 00:40:51.450 --> 00:40:53.750 align:middle line:90% Anyone have questions? 00:40:53.750 --> 00:40:55.800 align:middle line:90% Cool. 00:40:55.800 --> 00:40:58.490 align:middle line:84% Feel free to shout out questions if you have any, or interrupt. 00:40:58.490 --> 00:41:00.650 align:middle line:84% I'm more than happy to be interrupted. 00:41:00.650 --> 00:41:13.777 align:middle line:90% 00:41:13.777 --> 00:41:16.570 align:middle line:84% OK, so I'm just going to give one example of a metric space 00:41:16.570 --> 00:41:19.420 align:middle line:84% that is disconnected, which is one that makes relative sense, 00:41:19.420 --> 00:41:21.880 align:middle line:90% at least on the face of it. 00:41:21.880 --> 00:41:28.660 align:middle line:84% Example-- I can take a unit of two open intervals, 00:41:28.660 --> 00:41:44.390 align:middle line:84% CR to 1 and 1 to 2, and give it the usual metric on R. 00:41:44.390 --> 00:41:45.950 align:middle line:90% What do I mean by usual metric? 00:41:45.950 --> 00:41:48.650 align:middle line:84% I mean the distance between two points in the set 00:41:48.650 --> 00:41:50.840 align:middle line:84% is just the distance normally in R-- 00:41:50.840 --> 00:41:52.610 align:middle line:90% absolute values. 00:41:52.610 --> 00:41:55.040 align:middle line:84% This is an example of one that's disconnected, right? 00:41:55.040 --> 00:41:58.280 align:middle line:90% 00:41:58.280 --> 00:42:02.870 align:middle line:84% It is the union of two open sets that are disjoint. 00:42:02.870 --> 00:42:05.930 align:middle line:84% And you can notice here the interval 0, 00:42:05.930 --> 00:42:12.310 align:middle line:84% 1 is both open and closed because the complement of it 00:42:12.310 --> 00:42:14.540 align:middle line:90% is just 1, 2, and that's open. 00:42:14.540 --> 00:42:16.970 align:middle line:84% So this is an interesting example to think through. 00:42:16.970 --> 00:42:19.880 align:middle line:84% This is known as the subspace metric 00:42:19.880 --> 00:42:23.000 align:middle line:84% or the subspace topology, technically. 00:42:23.000 --> 00:42:24.510 align:middle line:90% But yeah. 00:42:24.510 --> 00:42:25.740 align:middle line:90% OK. 00:42:25.740 --> 00:42:28.110 align:middle line:84% Let's go back to proving things about the general theory 00:42:28.110 --> 00:42:30.420 align:middle line:90% of open sets. 00:42:30.420 --> 00:42:34.140 align:middle line:84% Proposition-- I'm first going to point out that closed sets are 00:42:34.140 --> 00:42:36.960 align:middle line:90% very similar to open sets. 00:42:36.960 --> 00:42:38.880 align:middle line:84% In fact, we can define everything 00:42:38.880 --> 00:42:41.817 align:middle line:84% in terms of closed sets, if we wanted to. 00:42:41.817 --> 00:42:42.900 align:middle line:90% Properties of closed sets. 00:42:42.900 --> 00:42:49.510 align:middle line:90% 00:42:49.510 --> 00:42:53.140 align:middle line:84% Here we have that the empty set in x are closed. 00:42:53.140 --> 00:42:55.810 align:middle line:90% 00:42:55.810 --> 00:43:04.390 align:middle line:84% Two, the potentially infinite intersection of closed sets, 00:43:04.390 --> 00:43:08.950 align:middle line:90% Ai closed, is closed. 00:43:08.950 --> 00:43:13.350 align:middle line:90% 00:43:13.350 --> 00:43:23.800 align:middle line:84% And three, the finite union of closed sets is closed. 00:43:23.800 --> 00:43:27.390 align:middle line:90% 00:43:27.390 --> 00:43:29.910 align:middle line:84% Nearly identical to the topological properties 00:43:29.910 --> 00:43:31.320 align:middle line:90% of open sets. 00:43:31.320 --> 00:43:32.850 align:middle line:90% How do we prove this? 00:43:32.850 --> 00:43:33.837 align:middle line:90% It's not too bad. 00:43:33.837 --> 00:43:35.170 align:middle line:90% I'm not going to actually do it. 00:43:35.170 --> 00:43:37.320 align:middle line:84% The fact that the empty set and x are closed 00:43:37.320 --> 00:43:40.680 align:middle line:84% is just the proof that I did above for R, right? 00:43:40.680 --> 00:43:44.470 align:middle line:84% You can replace R with just any metric space. 00:43:44.470 --> 00:43:46.720 align:middle line:84% And we have that this first property holds. 00:43:46.720 --> 00:43:48.130 align:middle line:90% How about the other two? 00:43:48.130 --> 00:43:49.930 align:middle line:84% Well, for the other two, I'm just 00:43:49.930 --> 00:43:52.660 align:middle line:84% going to quickly note what are known as De Morgan's laws. 00:43:52.660 --> 00:43:56.710 align:middle line:90% 00:43:56.710 --> 00:44:01.270 align:middle line:84% So this is a nice lemma from set theory. 00:44:01.270 --> 00:44:02.905 align:middle line:84% These are known as De Morgan's laws. 00:44:02.905 --> 00:44:08.510 align:middle line:90% 00:44:08.510 --> 00:44:10.260 align:middle line:84% I'm going to switch which chalk I'm using. 00:44:10.260 --> 00:44:13.440 align:middle line:90% 00:44:13.440 --> 00:44:21.210 align:middle line:84% This states that the complement of a union 00:44:21.210 --> 00:44:25.380 align:middle line:84% is the intersection of the complements-- 00:44:25.380 --> 00:44:35.900 align:middle line:84% so i am A. And similarly, the complement of an intersection-- 00:44:35.900 --> 00:44:40.310 align:middle line:90% Ui complement i and A-- 00:44:40.310 --> 00:44:42.650 align:middle line:84% is equal to the union of the complements. 00:44:42.650 --> 00:44:46.070 align:middle line:90% 00:44:46.070 --> 00:44:47.720 align:middle line:84% These are known as De Morgan's laws. 00:44:47.720 --> 00:44:48.820 align:middle line:90% They make relative sense? 00:44:48.820 --> 00:44:50.570 align:middle line:84% You can prove them for two sets as opposed 00:44:50.570 --> 00:44:52.260 align:middle line:90% to infinitely many of them. 00:44:52.260 --> 00:44:54.380 align:middle line:84% But once you have these two properties, 00:44:54.380 --> 00:44:56.480 align:middle line:84% then you can rewrite these two properties 00:44:56.480 --> 00:44:58.040 align:middle line:90% in terms of open sets. 00:44:58.040 --> 00:45:00.200 align:middle line:84% That's why there's this duality, right? 00:45:00.200 --> 00:45:02.090 align:middle line:84% I can write the intersection of closed sets 00:45:02.090 --> 00:45:03.455 align:middle line:90% as the union of open sets. 00:45:03.455 --> 00:45:07.300 align:middle line:90% 00:45:07.300 --> 00:45:08.948 align:middle line:84% That's how you apply De Morgan's laws. 00:45:08.948 --> 00:45:10.740 align:middle line:84% And I'm not going to prove that, or I'm not 00:45:10.740 --> 00:45:12.330 align:middle line:84% going to prove these topological properties. 00:45:12.330 --> 00:45:13.980 align:middle line:84% This is in the lecture notes, but I 00:45:13.980 --> 00:45:15.750 align:middle line:84% do want to note-- you could do everything 00:45:15.750 --> 00:45:18.600 align:middle line:84% in terms of closed sets if you really truly wanted to. 00:45:18.600 --> 00:45:23.280 align:middle line:84% We just don't because open sets are nice. 00:45:23.280 --> 00:45:24.210 align:middle line:90% OK. 00:45:24.210 --> 00:45:28.392 align:middle line:84% We'll show why some of those things are actually nice. 00:45:28.392 --> 00:45:29.850 align:middle line:84% What more things do I want to note? 00:45:29.850 --> 00:45:30.480 align:middle line:90% OK, cool. 00:45:30.480 --> 00:45:33.170 align:middle line:90% 00:45:33.170 --> 00:45:36.380 align:middle line:84% Now, I want to state some specific examples of open sets 00:45:36.380 --> 00:45:41.320 align:middle line:84% in metric spaces, which we can already do quite a bit of. 00:45:41.320 --> 00:45:45.698 align:middle line:84% Example-- did I erase it already? 00:45:45.698 --> 00:45:46.240 align:middle line:90% Oh, wait, no. 00:45:46.240 --> 00:45:46.960 align:middle line:90% I did it here. 00:45:46.960 --> 00:45:49.930 align:middle line:90% 00:45:49.930 --> 00:45:50.680 align:middle line:90% Where did I do it? 00:45:50.680 --> 00:45:52.570 align:middle line:90% Oh, I did at the very top. 00:45:52.570 --> 00:45:54.310 align:middle line:84% The ball of radius epsilon around A 00:45:54.310 --> 00:45:57.010 align:middle line:84% is sometimes referred to as an open ball, 00:45:57.010 --> 00:46:00.100 align:middle line:84% but you should prove that the ball is, in fact, open, which 00:46:00.100 --> 00:46:02.360 align:middle line:90% will be our first example. 00:46:02.360 --> 00:46:07.270 align:middle line:84% So given some point x in your metric space 00:46:07.270 --> 00:46:11.850 align:middle line:84% x and epsilon bigger than 0, we have 00:46:11.850 --> 00:46:15.465 align:middle line:84% that the ball of radius epsilon around x is open. 00:46:15.465 --> 00:46:18.040 align:middle line:90% 00:46:18.040 --> 00:46:18.950 align:middle line:90% How do we do this? 00:46:18.950 --> 00:46:20.440 align:middle line:84% Well, we choose an arbitrary point 00:46:20.440 --> 00:46:23.760 align:middle line:84% in your ball of radius epsilon, right? 00:46:23.760 --> 00:46:33.250 align:middle line:84% So proof-- choose some y in the ball 00:46:33.250 --> 00:46:36.040 align:middle line:90% of radius epsilon around x. 00:46:36.040 --> 00:46:38.080 align:middle line:84% We want to find a ball of radius, let's say, 00:46:38.080 --> 00:46:42.130 align:middle line:84% delta around y contained in the ball of radius epsilon round x. 00:46:42.130 --> 00:46:45.940 align:middle line:84% That's the definition of open set. 00:46:45.940 --> 00:46:50.920 align:middle line:84% Well, notice that the distance from x to y 00:46:50.920 --> 00:46:53.500 align:middle line:84% is less than epsilon by definition, right? 00:46:53.500 --> 00:46:55.180 align:middle line:84% Because we're assuming that the distance 00:46:55.180 --> 00:46:58.610 align:middle line:84% between points in our ball is less than epsilon. 00:46:58.610 --> 00:47:08.350 align:middle line:84% So let delta be the distance from x to y over 2. 00:47:08.350 --> 00:47:09.470 align:middle line:90% Why does this work? 00:47:09.470 --> 00:47:12.700 align:middle line:84% Because then we'll notice that the ball 00:47:12.700 --> 00:47:16.030 align:middle line:84% of radius delta around y has to be 00:47:16.030 --> 00:47:19.180 align:middle line:84% contained in the ball of radius epsilon around x. 00:47:19.180 --> 00:47:20.590 align:middle line:90% Why is this true? 00:47:20.590 --> 00:47:23.140 align:middle line:84% Essentially, the triangle inequality, right? 00:47:23.140 --> 00:47:30.310 align:middle line:84% We have a ball of radius epsilon centered at x. 00:47:30.310 --> 00:47:33.560 align:middle line:90% We have a point y here. 00:47:33.560 --> 00:47:35.840 align:middle line:84% And I'm noting that the distance from here to here 00:47:35.840 --> 00:47:41.250 align:middle line:84% is less than epsilon, by definition. 00:47:41.250 --> 00:47:44.420 align:middle line:84% And I'm just choosing a ball around y 00:47:44.420 --> 00:47:46.823 align:middle line:90% to be 1/2 of that distance. 00:47:46.823 --> 00:47:48.740 align:middle line:84% And then you can apply the triangle inequality 00:47:48.740 --> 00:47:50.600 align:middle line:84% to show that this ball must actually 00:47:50.600 --> 00:47:52.900 align:middle line:90% be contained in the bigger one. 00:47:52.900 --> 00:47:54.400 align:middle line:84% Anyone want me to work through that? 00:47:54.400 --> 00:47:55.220 align:middle line:90% I'm happy to. 00:47:55.220 --> 00:48:00.020 align:middle line:90% 00:48:00.020 --> 00:48:01.050 align:middle line:90% Oh, wait a moment. 00:48:01.050 --> 00:48:08.660 align:middle line:90% 00:48:08.660 --> 00:48:10.912 align:middle line:90% Let me write it up. 00:48:10.912 --> 00:48:11.495 align:middle line:90% You know what? 00:48:11.495 --> 00:48:13.990 align:middle line:84% I'll write it out on the back board. 00:48:13.990 --> 00:48:21.770 align:middle line:84% So let z be in the ball of radius delta around y. 00:48:21.770 --> 00:48:26.290 align:middle line:84% Then we have that the distance from x to z 00:48:26.290 --> 00:48:29.630 align:middle line:84% is less than or equal to, by the triangle inequality, 00:48:29.630 --> 00:48:36.160 align:middle line:84% the distance from x to y plus the distance from y to z-- 00:48:36.160 --> 00:48:39.430 align:middle line:90% distance from x to y. 00:48:39.430 --> 00:48:42.040 align:middle line:84% And the distance from y to z is less than the distance 00:48:42.040 --> 00:48:43.240 align:middle line:90% from x to y over 2. 00:48:43.240 --> 00:48:49.070 align:middle line:90% 00:48:49.070 --> 00:48:50.450 align:middle line:90% Is this what I wanted to do? 00:48:50.450 --> 00:48:56.710 align:middle line:90% 00:48:56.710 --> 00:48:58.450 align:middle line:90% Oh, I should say, no. 00:48:58.450 --> 00:49:01.690 align:middle line:90% This is not what I wanted to do. 00:49:01.690 --> 00:49:10.900 align:middle line:84% Let R be the distance from x to y minus epsilon. 00:49:10.900 --> 00:49:13.510 align:middle line:84% Then what we know is that the distance from y to z 00:49:13.510 --> 00:49:19.750 align:middle line:84% is less than R. What am I trying to say? 00:49:19.750 --> 00:49:22.840 align:middle line:84% Distance from y to z is less than-- 00:49:22.840 --> 00:49:25.210 align:middle line:84% or sorry, this should be epsilon minus the distance. 00:49:25.210 --> 00:49:25.930 align:middle line:90% There we go. 00:49:25.930 --> 00:49:28.280 align:middle line:84% And this is then less than epsilon. 00:49:28.280 --> 00:49:29.650 align:middle line:90% So this is in the lecture notes. 00:49:29.650 --> 00:49:31.690 align:middle line:84% I would highly suggest drawing up the picture 00:49:31.690 --> 00:49:34.150 align:middle line:84% and working through it separately. 00:49:34.150 --> 00:49:36.762 align:middle line:84% But loosely speaking, it's just this diagram 00:49:36.762 --> 00:49:37.720 align:middle line:90% on the right-hand side. 00:49:37.720 --> 00:49:41.680 align:middle line:84% It's an application of the triangle inequality. 00:49:41.680 --> 00:49:42.180 align:middle line:90% OK. 00:49:42.180 --> 00:49:45.150 align:middle line:84% So the ball of radius epsilon is an open ball. 00:49:45.150 --> 00:49:48.840 align:middle line:84% In fact, you can state much more than this. 00:49:48.840 --> 00:49:53.150 align:middle line:84% This is an optional problem on your second problem set. 00:49:53.150 --> 00:49:58.933 align:middle line:84% And for the record, I do suggest y'all look at the optional 00:49:58.933 --> 00:50:01.100 align:middle line:84% problems because they will potentially give you more 00:50:01.100 --> 00:50:02.900 align:middle line:84% intuition, even if you don't solve them. 00:50:02.900 --> 00:50:06.310 align:middle line:90% 00:50:06.310 --> 00:50:10.420 align:middle line:84% But yeah, this is just one of those cases where this 00:50:10.420 --> 00:50:13.360 align:middle line:84% isn't a topology class, so I'm not going to ask that y'all do 00:50:13.360 --> 00:50:18.240 align:middle line:84% a bunch of topology, but it's pretty nice. 00:50:18.240 --> 00:50:19.630 align:middle line:90% OK. 00:50:19.630 --> 00:50:21.280 align:middle line:90% So just a small note. 00:50:21.280 --> 00:50:25.420 align:middle line:90% 00:50:25.420 --> 00:50:33.550 align:middle line:84% Any open set U contained in your metric space 00:50:33.550 --> 00:50:44.770 align:middle line:84% can be written as a union, a potentially infinite union, 00:50:44.770 --> 00:50:45.640 align:middle line:90% of open balls. 00:50:45.640 --> 00:50:51.490 align:middle line:90% 00:50:51.490 --> 00:50:53.660 align:middle line:84% I'll outline briefly how to do this right now. 00:50:53.660 --> 00:50:56.360 align:middle line:84% So if y'all want to do it later, you can do it. 00:50:56.360 --> 00:50:59.470 align:middle line:84% But the idea is that for any single point in your open set, 00:50:59.470 --> 00:51:02.578 align:middle line:84% there exists a ball of radius epsilon around that point. 00:51:02.578 --> 00:51:04.870 align:middle line:84% And then you can just take the union of all those balls 00:51:04.870 --> 00:51:05.870 align:middle line:90% of radius epsilon. 00:51:05.870 --> 00:51:07.870 align:middle line:84% The epsilon might change for every single point, 00:51:07.870 --> 00:51:09.162 align:middle line:90% but that doesn't matter, right? 00:51:09.162 --> 00:51:10.660 align:middle line:90% Because it's an infinite union. 00:51:10.660 --> 00:51:13.780 align:middle line:84% So I'll let y'all work through those details, 00:51:13.780 --> 00:51:17.995 align:middle line:84% but this is just a proposition you can show. 00:51:17.995 --> 00:51:20.070 align:middle line:84% OK, let's also just prove an example 00:51:20.070 --> 00:51:25.313 align:middle line:84% of a closed set, which will come important for next time. 00:51:25.313 --> 00:51:25.980 align:middle line:90% Just an example. 00:51:25.980 --> 00:51:28.870 align:middle line:90% 00:51:28.870 --> 00:51:29.370 align:middle line:90% Oh, sorry. 00:51:29.370 --> 00:51:35.850 align:middle line:84% I should say let x be a point in your metric space. 00:51:35.850 --> 00:51:40.680 align:middle line:90% Then the set x is closed. 00:51:40.680 --> 00:51:44.910 align:middle line:84% So this is just a singular point in your metric space. 00:51:44.910 --> 00:51:46.210 align:middle line:90% How are we going to prove this? 00:51:46.210 --> 00:51:48.060 align:middle line:84% Well, we only have one definition of closed. 00:51:48.060 --> 00:51:53.790 align:middle line:84% We have that the complement is open, right? 00:51:53.790 --> 00:52:01.020 align:middle line:84% So proof-- consider the complement 00:52:01.020 --> 00:52:15.330 align:middle line:84% of x, which is x not including x, and let y be a point here. 00:52:15.330 --> 00:52:17.520 align:middle line:84% What we want to show is that there's 00:52:17.520 --> 00:52:19.180 align:middle line:90% a ball of radius epsilon-- 00:52:19.180 --> 00:52:23.130 align:middle line:90% want to show-- around y-- 00:52:23.130 --> 00:52:28.520 align:middle line:84% or not in-- a subset of x not including x. 00:52:28.520 --> 00:52:31.020 align:middle line:84% Let's draw a picture of what this should actually look like, 00:52:31.020 --> 00:52:33.650 align:middle line:90% right? 00:52:33.650 --> 00:52:35.650 align:middle line:90% Well, I didn't want to do that. 00:52:35.650 --> 00:52:38.530 align:middle line:90% Here's our metric space. 00:52:38.530 --> 00:52:40.600 align:middle line:90% Here's our singular point x. 00:52:40.600 --> 00:52:43.420 align:middle line:84% And here's our point, let's say, y. 00:52:43.420 --> 00:52:48.940 align:middle line:84% We want to choose a ball around y that doesn't contain x. 00:52:48.940 --> 00:52:49.750 align:middle line:90% How can we do it? 00:52:49.750 --> 00:52:52.990 align:middle line:90% 00:52:52.990 --> 00:52:53.628 align:middle line:90% Anyone? 00:52:53.628 --> 00:52:55.420 align:middle line:84% You could choose whatever epsilon you want, 00:52:55.420 --> 00:52:57.670 align:middle line:84% so it's a question of what epsilon you want to choose. 00:52:57.670 --> 00:53:00.570 align:middle line:90% 00:53:00.570 --> 00:53:04.158 align:middle line:84% We can choose epsilon to be 1/2 the distance between x and y. 00:53:04.158 --> 00:53:06.450 align:middle line:84% I think that's what I was mixing up on the other proof, 00:53:06.450 --> 00:53:08.970 align:middle line:84% but it's in the lecture notes, so I'll let y'all do the other 00:53:08.970 --> 00:53:09.470 align:middle line:90% one. 00:53:09.470 --> 00:53:20.220 align:middle line:84% But choose epsilon to be the distance from x to y over 2. 00:53:20.220 --> 00:53:24.120 align:middle line:84% And then pictorially, we have the rest of the problem, 00:53:24.120 --> 00:53:26.810 align:middle line:84% but let's actually work through the details. 00:53:26.810 --> 00:53:32.850 align:middle line:84% So then let z be in your ball of radius epsilon around y. 00:53:32.850 --> 00:53:38.930 align:middle line:84% Then we want to show that z cannot be x, right? 00:53:38.930 --> 00:53:45.210 align:middle line:84% Well, notice-- this implies that the distance from y to z 00:53:45.210 --> 00:53:50.160 align:middle line:84% has to be less than epsilon, right? 00:53:50.160 --> 00:53:55.050 align:middle line:84% This is by construction, but can we get a lower bound on this? 00:53:55.050 --> 00:53:56.376 align:middle line:90% We can. 00:53:56.376 --> 00:53:59.468 align:middle line:84% We can note that the distance from-- 00:53:59.468 --> 00:54:01.260 align:middle line:84% I want to make sure I'm getting this right. 00:54:01.260 --> 00:54:10.330 align:middle line:90% 00:54:10.330 --> 00:54:11.330 align:middle line:90% Actually, you know what? 00:54:11.330 --> 00:54:13.663 align:middle line:84% I'll do this directly, or I'll do this by contradiction. 00:54:13.663 --> 00:54:16.480 align:middle line:90% 00:54:16.480 --> 00:54:19.770 align:middle line:84% Suppose, for the sake of contradiction, 00:54:19.770 --> 00:54:22.540 align:middle line:84% x was in the ball of radius epsilon around y. 00:54:22.540 --> 00:54:26.130 align:middle line:84% Then this would imply that the distance from x to y 00:54:26.130 --> 00:54:29.820 align:middle line:84% is less than epsilon, but this can't be the case 00:54:29.820 --> 00:54:35.760 align:middle line:84% because again, we're assuming that epsilon 00:54:35.760 --> 00:54:38.470 align:middle line:84% is the distance from x to y over 2. 00:54:38.470 --> 00:54:40.470 align:middle line:84% And the distance from x to y cannot be less than 00:54:40.470 --> 00:54:42.780 align:middle line:90% the distance from x to y over 2. 00:54:42.780 --> 00:54:44.926 align:middle line:90% So that is our contradiction. 00:54:44.926 --> 00:54:49.890 align:middle line:90% 00:54:49.890 --> 00:54:51.880 align:middle line:84% I'll make one more small note here, 00:54:51.880 --> 00:54:55.080 align:middle line:84% which will become important for Tuesday, which is, as opposed 00:54:55.080 --> 00:54:56.940 align:middle line:84% to looking at a singular point, you 00:54:56.940 --> 00:55:01.218 align:middle line:84% can do this with a finite union of points, right? 00:55:01.218 --> 00:55:02.760 align:middle line:84% And the argument would be essentially 00:55:02.760 --> 00:55:05.130 align:middle line:84% the same as our finite intersection of open sets 00:55:05.130 --> 00:55:06.060 align:middle line:90% argument. 00:55:06.060 --> 00:55:09.430 align:middle line:84% You have a finite Union of open sets-- 00:55:09.430 --> 00:55:11.180 align:middle line:84% or sorry, a finite union of closed things. 00:55:11.180 --> 00:55:12.220 align:middle line:90% So it should be closed. 00:55:12.220 --> 00:55:14.840 align:middle line:84% In fact, you could use that to prove it immediately. 00:55:14.840 --> 00:55:18.972 align:middle line:84% But you can also prove it directly, like this. 00:55:18.972 --> 00:55:21.555 align:middle line:90% OK. 00:55:21.555 --> 00:55:22.930 align:middle line:84% So now, what we're going to do is 00:55:22.930 --> 00:55:26.950 align:middle line:84% talk about how this definition of open sets 00:55:26.950 --> 00:55:34.960 align:middle line:84% relates to convergent sequences and to continuity. 00:55:34.960 --> 00:55:40.155 align:middle line:90% Proposition-- let xn-- 00:55:40.155 --> 00:55:42.280 align:middle line:84% to do this, I'm going to do it in the real numbers, 00:55:42.280 --> 00:55:44.697 align:middle line:84% but note that everything can be done in your metric space. 00:55:44.697 --> 00:55:48.170 align:middle line:84% I just want to use the notation from the real numbers. 00:55:48.170 --> 00:55:52.390 align:middle line:90% Let xn be a sequence-- 00:55:52.390 --> 00:55:53.530 align:middle line:90% am I writing it backwards? 00:55:53.530 --> 00:56:01.140 align:middle line:90% I am-- in R that converges to x. 00:56:01.140 --> 00:56:04.260 align:middle line:90% 00:56:04.260 --> 00:56:06.750 align:middle line:84% So then this is true, I claim, if and only 00:56:06.750 --> 00:56:24.210 align:middle line:84% if, for all epsilon bigger than 0, all but finitely many xi 00:56:24.210 --> 00:56:32.030 align:middle line:84% are in x minus epsilon to x plus epsilon. 00:56:32.030 --> 00:56:36.100 align:middle line:84% So this is the ball of radius epsilon around x, right? 00:56:36.100 --> 00:56:38.350 align:middle line:84% How do we prove this if and only if? 00:56:38.350 --> 00:56:41.230 align:middle line:90% Proof-- well, OK. 00:56:41.230 --> 00:56:44.500 align:middle line:84% I should say convergent sequence. 00:56:44.500 --> 00:56:46.220 align:middle line:84% Let's prove the forward direction first. 00:56:46.220 --> 00:56:54.070 align:middle line:84% So if xn is convergent, then for all epsilon bigger than 0, 00:56:54.070 --> 00:56:59.680 align:middle line:84% there exists n in the natural numbers such that the ball 00:56:59.680 --> 00:57:00.985 align:middle line:90% of radius epsilon around x-- 00:57:00.985 --> 00:57:04.090 align:middle line:90% 00:57:04.090 --> 00:57:11.710 align:middle line:84% I should say such that for all n bigger than or equal to n, 00:57:11.710 --> 00:57:20.860 align:middle line:84% xn is in the ball of radius epsilon, x plus epsilon. 00:57:20.860 --> 00:57:24.280 align:middle line:84% This is true because the distance from xn to x 00:57:24.280 --> 00:57:25.400 align:middle line:90% is less than epsilon. 00:57:25.400 --> 00:57:29.050 align:middle line:84% This is just the open set way to view it, right? 00:57:29.050 --> 00:57:30.550 align:middle line:90% And this capital N-- 00:57:30.550 --> 00:57:32.920 align:middle line:84% notice that there are only finitely many terms less 00:57:32.920 --> 00:57:33.550 align:middle line:90% than it. 00:57:33.550 --> 00:57:35.560 align:middle line:84% So this shows that infinitely many of the terms 00:57:35.560 --> 00:57:36.820 align:middle line:90% are in this open set. 00:57:36.820 --> 00:57:39.730 align:middle line:84% And in fact, only finitely many are outside of it. 00:57:39.730 --> 00:57:41.620 align:middle line:84% So that proves the forward direction. 00:57:41.620 --> 00:57:43.240 align:middle line:84% Let's prove the opposite direction. 00:57:43.240 --> 00:57:52.140 align:middle line:90% 00:57:52.140 --> 00:58:00.464 align:middle line:84% I should write-- suppose, for all epsilon bigger than 0, 00:58:00.464 --> 00:58:14.680 align:middle line:84% all but finitely many xi are in x minus epsilon 00:58:14.680 --> 00:58:17.440 align:middle line:90% to x plus epsilon. 00:58:17.440 --> 00:58:22.970 align:middle line:84% I want to choose a sequence that will converge to x then, right? 00:58:22.970 --> 00:58:25.170 align:middle line:90% Well, how can we do that? 00:58:25.170 --> 00:58:28.430 align:middle line:84% We can construct a subsequence that is going to be convergent. 00:58:28.430 --> 00:58:32.240 align:middle line:84% And remember, a subsequence, if it's convergent, 00:58:32.240 --> 00:58:36.212 align:middle line:84% converges to what the original sequence converges to, right? 00:58:36.212 --> 00:58:37.670 align:middle line:84% So this is where we're using a fact 00:58:37.670 --> 00:58:40.390 align:middle line:90% from earlier that we proved. 00:58:40.390 --> 00:58:41.320 align:middle line:90% OK. 00:58:41.320 --> 00:58:46.750 align:middle line:84% So suppose epsilon is 1 over n for n in the natural numbers, 00:58:46.750 --> 00:58:49.840 align:middle line:90% or I'll say m. 00:58:49.840 --> 00:59:00.680 align:middle line:84% Then choose x and m in the ball of radius epsilon-- 00:59:00.680 --> 00:59:05.600 align:middle line:90% so 1 over m to x plus 1 over m. 00:59:05.600 --> 00:59:06.980 align:middle line:90% What does this tell us? 00:59:06.980 --> 00:59:12.050 align:middle line:84% I claim that this convergent sequence converges to x. 00:59:12.050 --> 00:59:18.020 align:middle line:84% Well, to show that, we need to write down the definition. 00:59:18.020 --> 00:59:25.010 align:middle line:90% 00:59:25.010 --> 00:59:27.530 align:middle line:84% When Professor Rodriguez was erasing the board, 00:59:27.530 --> 00:59:30.170 align:middle line:84% he'd always tell some joke, but I don't have any jokes 00:59:30.170 --> 00:59:31.700 align:middle line:90% right now, unfortunately. 00:59:31.700 --> 00:59:35.100 align:middle line:90% 00:59:35.100 --> 00:59:36.850 align:middle line:84% Though he would do them two at a time-- so 00:59:36.850 --> 00:59:38.642 align:middle line:84% he could actually have time to actually say 00:59:38.642 --> 00:59:40.894 align:middle line:90% the joke before he finished. 00:59:40.894 --> 00:59:41.760 align:middle line:90% OK. 00:59:41.760 --> 00:59:45.702 align:middle line:84% So we want to show that x and m converges to x, right? 00:59:45.702 --> 00:59:49.510 align:middle line:84% Proof-- for all epsilon bigger than 0, 00:59:49.510 --> 00:59:55.165 align:middle line:90% we can choose M large enough-- 00:59:55.165 --> 00:59:59.750 align:middle line:90% 00:59:59.750 --> 01:00:02.210 align:middle line:90% or I should say capital M-- 01:00:02.210 --> 01:00:08.170 align:middle line:84% such that for all nm bigger than or equal to M, 01:00:08.170 --> 01:00:16.000 align:middle line:84% we have that 1 over nm is less than epsilon, right? 01:00:16.000 --> 01:00:17.680 align:middle line:84% Because epsilon is bigger than 0, 01:00:17.680 --> 01:00:19.810 align:middle line:84% I can find a large enough natural number such 01:00:19.810 --> 01:00:22.480 align:middle line:84% that 1 over that natural number is less than epsilon. 01:00:22.480 --> 01:00:26.870 align:middle line:84% That's a fact from real analysis. 01:00:26.870 --> 01:00:34.360 align:middle line:84% So what this tells us is that we know, then, 01:00:34.360 --> 01:00:46.160 align:middle line:84% that x and M is in x minus epsilon to x plus epsilon 01:00:46.160 --> 01:00:53.690 align:middle line:84% because this is from x over 1 over nm to x plus 1 over nm, 01:00:53.690 --> 01:00:56.960 align:middle line:84% which is a subset of the ball of radius epsilon, 01:00:56.960 --> 01:00:59.180 align:middle line:90% by construction, right? 01:00:59.180 --> 01:01:01.290 align:middle line:84% Because 1 over nm is less than epsilon. 01:01:01.290 --> 01:01:06.440 align:middle line:84% So what this tells you is that the distance from x nm minus x 01:01:06.440 --> 01:01:08.510 align:middle line:84% is less than epsilon, which implies 01:01:08.510 --> 01:01:11.480 align:middle line:84% that the convergent sequence or the convergent subsequence 01:01:11.480 --> 01:01:12.830 align:middle line:90% converges to x. 01:01:12.830 --> 01:01:14.780 align:middle line:84% So to reiterate how the proof went-- 01:01:14.780 --> 01:01:16.310 align:middle line:90% we had a convergent sequence. 01:01:16.310 --> 01:01:19.190 align:middle line:84% We had a subsequence that we showed converges to x. 01:01:19.190 --> 01:01:21.327 align:middle line:84% And by what we showed earlier, x must 01:01:21.327 --> 01:01:23.660 align:middle line:84% be the limit point of the original convergence sequence, 01:01:23.660 --> 01:01:24.410 align:middle line:90% then. 01:01:24.410 --> 01:01:25.310 align:middle line:90% So we're done. 01:01:25.310 --> 01:01:29.140 align:middle line:90% 01:01:29.140 --> 01:01:31.070 align:middle line:90% All right. 01:01:31.070 --> 01:01:34.010 align:middle line:84% I won't write out the analogous statement for metric spaces. 01:01:34.010 --> 01:01:35.900 align:middle line:84% You just change x minus epsilon to x 01:01:35.900 --> 01:01:38.750 align:middle line:84% plus epsilon with the ball of radius epsilon. 01:01:38.750 --> 01:01:42.740 align:middle line:84% I just stated it in real numbers because I 01:01:42.740 --> 01:01:44.930 align:middle line:84% prefer the interval notation to think about it. 01:01:44.930 --> 01:01:47.108 align:middle line:84% And it's a fact that not everyone fully 01:01:47.108 --> 01:01:49.400 align:middle line:84% sees in real analysis because sometimes, you don't know 01:01:49.400 --> 01:01:51.650 align:middle line:90% what an open set is in 100A. 01:01:51.650 --> 01:01:54.940 align:middle line:84% It depends on who's teaching it, realistically. 01:01:54.940 --> 01:01:56.050 align:middle line:90% OK. 01:01:56.050 --> 01:01:58.660 align:middle line:90% We're going to prove one more-- 01:01:58.660 --> 01:01:59.160 align:middle line:90% oh, wait. 01:01:59.160 --> 01:02:00.077 align:middle line:90% What do I want to say? 01:02:00.077 --> 01:02:03.870 align:middle line:90% 01:02:03.870 --> 01:02:05.063 align:middle line:90% Yeah, OK. 01:02:05.063 --> 01:02:07.230 align:middle line:84% This was just how I wanted to show that it's related 01:02:07.230 --> 01:02:09.570 align:middle line:90% to convergent sequences. 01:02:09.570 --> 01:02:11.370 align:middle line:84% Now, what I'm going to do is show 01:02:11.370 --> 01:02:14.422 align:middle line:84% that it's related to continuous functions. 01:02:14.422 --> 01:02:16.380 align:middle line:84% And I'm going to recall what that definition is 01:02:16.380 --> 01:02:18.047 align:middle line:84% because that can, again, be a little bit 01:02:18.047 --> 01:02:20.260 align:middle line:90% weird between metric spaces. 01:02:20.260 --> 01:02:22.770 align:middle line:84% So a function f from a metric space 01:02:22.770 --> 01:02:28.890 align:middle line:84% x to a metric space y, let's say, with metric dx and dy, 01:02:28.890 --> 01:02:37.430 align:middle line:84% is continuous if, for all epsilon bigger than 0, 01:02:37.430 --> 01:02:41.510 align:middle line:84% there exists a delta bigger than 0 such 01:02:41.510 --> 01:02:46.550 align:middle line:84% that if the distance between two points x and y 01:02:46.550 --> 01:02:49.790 align:middle line:84% is less than delta, then the distance in the metric space 01:02:49.790 --> 01:02:55.520 align:middle line:84% y of the images of x and y is less than epsilon. 01:02:55.520 --> 01:02:58.450 align:middle line:84% This is the definition of continuous in the metric space 01:02:58.450 --> 01:03:00.430 align:middle line:90% setting. 01:03:00.430 --> 01:03:01.950 align:middle line:84% And so now what we're going to do 01:03:01.950 --> 01:03:05.310 align:middle line:84% is prove a statement akin to convergent sequences, 01:03:05.310 --> 01:03:06.390 align:middle line:90% but for continuity. 01:03:06.390 --> 01:03:08.670 align:middle line:84% We can write the definition of continuity 01:03:08.670 --> 01:03:10.200 align:middle line:90% in terms of open sets. 01:03:10.200 --> 01:03:13.350 align:middle line:84% And not only can we do that, we can write continuity in terms 01:03:13.350 --> 01:03:14.997 align:middle line:90% of convergence of sequences. 01:03:14.997 --> 01:03:16.080 align:middle line:90% So I'll do that right now. 01:03:16.080 --> 01:03:19.580 align:middle line:90% 01:03:19.580 --> 01:03:24.110 align:middle line:84% Theorem-- under all the same assumptions here 01:03:24.110 --> 01:03:27.230 align:middle line:84% I'm going to let x and y be metric spaces with distances dx 01:03:27.230 --> 01:03:29.060 align:middle line:90% and dy. 01:03:29.060 --> 01:03:33.360 align:middle line:90% And yeah, OK. 01:03:33.360 --> 01:03:37.950 align:middle line:90% 01:03:37.950 --> 01:03:43.530 align:middle line:84% Claim-- f from x to y is continuous at a point 01:03:43.530 --> 01:04:01.440 align:middle line:84% c in x if and only if xn is a sequence that converges to x. 01:04:01.440 --> 01:04:06.090 align:middle line:84% Then f of xn converges to f of x. 01:04:06.090 --> 01:04:08.970 align:middle line:90% This is my statement. 01:04:08.970 --> 01:04:12.030 align:middle line:84% When I say continuous at the point c, 01:04:12.030 --> 01:04:16.410 align:middle line:84% I mean just replace y with c here. 01:04:16.410 --> 01:04:21.540 align:middle line:84% That means that this is continuous at every point, 01:04:21.540 --> 01:04:24.810 align:middle line:90% continuous at x in x. 01:04:24.810 --> 01:04:28.328 align:middle line:84% And it's continuous if it's continuous everywhere, right? 01:04:28.328 --> 01:04:29.870 align:middle line:84% This is the same as in real analysis. 01:04:29.870 --> 01:04:32.570 align:middle line:84% We had continuity at a point and continuity everywhere. 01:04:32.570 --> 01:04:35.450 align:middle line:84% This is just continuity at a point. 01:04:35.450 --> 01:04:36.410 align:middle line:90% OK. 01:04:36.410 --> 01:04:43.100 align:middle line:84% Proof-- I always prefer going the downwards direction first. 01:04:43.100 --> 01:04:50.695 align:middle line:90% So let f be continuous at c. 01:04:50.695 --> 01:04:56.620 align:middle line:90% 01:04:56.620 --> 01:04:59.380 align:middle line:84% Then by definition, for all epsilon bigger than 0, 01:04:59.380 --> 01:05:02.860 align:middle line:84% there exists a delta such that this implication holds. 01:05:02.860 --> 01:05:05.050 align:middle line:90% What does this actually tell us? 01:05:05.050 --> 01:05:13.570 align:middle line:84% Well, I'm going to suppose the thing in the second hypothesis. 01:05:13.570 --> 01:05:20.440 align:middle line:84% Suppose xn is a sequence in x that converges to x. 01:05:20.440 --> 01:05:22.190 align:middle line:90% Then what can I say? 01:05:22.190 --> 01:05:25.473 align:middle line:84% Well, we know that there exists-- 01:05:25.473 --> 01:05:26.640 align:middle line:90% or sorry, I should say to c. 01:05:26.640 --> 01:05:29.930 align:middle line:90% 01:05:29.930 --> 01:05:32.790 align:middle line:84% Then by definition of convergent sequence, 01:05:32.790 --> 01:05:35.930 align:middle line:84% we know that for all epsilon bigger than 0, 01:05:35.930 --> 01:05:39.230 align:middle line:84% there exists an n in the natural numbers 01:05:39.230 --> 01:05:44.420 align:middle line:84% such that for all n bigger than or equal to n, 01:05:44.420 --> 01:05:48.660 align:middle line:84% the distance from xn to c is less than epsilon. 01:05:48.660 --> 01:05:57.880 align:middle line:84% But we know that for all epsilon bigger than 0, furthermore, 01:05:57.880 --> 01:06:00.490 align:middle line:84% there exists a delta bigger than 0 such 01:06:00.490 --> 01:06:07.240 align:middle line:84% that the distance from f of xn to f of c is less than delta 01:06:07.240 --> 01:06:08.170 align:middle line:90% over 2-- 01:06:08.170 --> 01:06:11.560 align:middle line:84% or sorry, not delta over 2, just delta. 01:06:11.560 --> 01:06:13.112 align:middle line:90% No. 01:06:13.112 --> 01:06:14.320 align:middle line:90% I should have relabeled this. 01:06:14.320 --> 01:06:16.278 align:middle line:84% I should have said for all delta bigger than 0, 01:06:16.278 --> 01:06:18.640 align:middle line:84% there exists one such that this is less than delta. 01:06:18.640 --> 01:06:22.520 align:middle line:84% And then there exists an epsilon bigger than 0 such that-- 01:06:22.520 --> 01:06:23.020 align:middle line:90% no, no. 01:06:23.020 --> 01:06:25.280 align:middle line:90% I said it right the first time. 01:06:25.280 --> 01:06:28.100 align:middle line:84% For all epsiolon bigger than 0, there exists a delta bigger 01:06:28.100 --> 01:06:30.590 align:middle line:84% than 0 such that this term is less than delta. 01:06:30.590 --> 01:06:34.100 align:middle line:84% And furthermore, the distance between the images 01:06:34.100 --> 01:06:35.190 align:middle line:90% is less than epsilon. 01:06:35.190 --> 01:06:35.700 align:middle line:90% There we go. 01:06:35.700 --> 01:06:37.160 align:middle line:90% That's what I meant to say. 01:06:37.160 --> 01:06:39.620 align:middle line:84% And so what this tells you is that f of xn 01:06:39.620 --> 01:06:41.870 align:middle line:90% must converge to f of c, right? 01:06:41.870 --> 01:06:43.970 align:middle line:84% Because there exists an n in the natural numbers 01:06:43.970 --> 01:06:45.980 align:middle line:84% such that the distance between the two points 01:06:45.980 --> 01:06:48.260 align:middle line:84% is less than epsilon, which is the definition 01:06:48.260 --> 01:06:50.780 align:middle line:90% of convergent sequence. 01:06:50.780 --> 01:06:53.360 align:middle line:84% And then to prove the other direction, 01:06:53.360 --> 01:06:54.980 align:middle line:84% I'm just going to state it because-- 01:06:54.980 --> 01:06:56.210 align:middle line:90% are we running out of time? 01:06:56.210 --> 01:06:58.193 align:middle line:90% No, I'll write it out. 01:06:58.193 --> 01:06:59.360 align:middle line:90% I'll write it out over here. 01:06:59.360 --> 01:07:03.052 align:middle line:90% 01:07:03.052 --> 01:07:04.510 align:middle line:84% To prove the other direction, we're 01:07:04.510 --> 01:07:07.180 align:middle line:84% going to use it by contradiction. 01:07:07.180 --> 01:07:10.750 align:middle line:84% We're going to assume that it's not continuous at c, 01:07:10.750 --> 01:07:17.230 align:middle line:84% and then choose a sequence that doesn't converge, 01:07:17.230 --> 01:07:18.880 align:middle line:84% or such that the images don't converge. 01:07:18.880 --> 01:07:27.150 align:middle line:90% 01:07:27.150 --> 01:07:29.180 align:middle line:90% OK. 01:07:29.180 --> 01:07:31.070 align:middle line:90% If and only if. 01:07:31.070 --> 01:07:38.060 align:middle line:90% Suppose f is not continuous. 01:07:38.060 --> 01:07:42.530 align:middle line:84% This is the proof of the upwards direction. 01:07:42.530 --> 01:07:45.920 align:middle line:84% Suppose it's not continuous at c. 01:07:45.920 --> 01:07:47.480 align:middle line:90% What does this tell you? 01:07:47.480 --> 01:07:50.570 align:middle line:84% Well, saying something is not continuous this is a little bit 01:07:50.570 --> 01:07:53.090 align:middle line:84% weird, but I'll state out what that definition means. 01:07:53.090 --> 01:07:58.640 align:middle line:84% Given some epsilon bigger than 0, 01:07:58.640 --> 01:08:06.890 align:middle line:84% we know that there exists an xn in the sequence 01:08:06.890 --> 01:08:13.220 align:middle line:84% such that the distance from xn to c gets small. 01:08:13.220 --> 01:08:16.270 align:middle line:90% 01:08:16.270 --> 01:08:17.270 align:middle line:90% Do I want to say over n? 01:08:17.270 --> 01:08:22.127 align:middle line:90% No, I'll just say less than-- 01:08:22.127 --> 01:08:24.460 align:middle line:84% given epsilon bigger than 0, there exists a delta bigger 01:08:24.460 --> 01:08:27.790 align:middle line:84% than 0 in an xn in your sequence such that the distances get 01:08:27.790 --> 01:08:28.490 align:middle line:90% small-- 01:08:28.490 --> 01:08:31.819 align:middle line:90% so let's say 1 over n. 01:08:31.819 --> 01:08:38.630 align:middle line:84% But the distances between the images of them, f of c, 01:08:38.630 --> 01:08:41.319 align:middle line:90% is bigger than epsilon, right? 01:08:41.319 --> 01:08:47.020 align:middle line:84% Because the opposite would be if this was less than 01:08:47.020 --> 01:08:50.439 align:middle line:90% or equal to epsilon, right? 01:08:50.439 --> 01:08:56.260 align:middle line:84% And so then what you can show is that therefore, xn 01:08:56.260 --> 01:09:04.960 align:middle line:84% converges to c, but f of xn does not converge to f of c. 01:09:04.960 --> 01:09:07.600 align:middle line:84% How do we know that it doesn't converge to f of c? 01:09:07.600 --> 01:09:12.640 align:middle line:84% Because we just proved up here that if a sequence is 01:09:12.640 --> 01:09:15.490 align:middle line:84% to converge all but finitely many of them, i.e. 01:09:15.490 --> 01:09:17.170 align:middle line:90% infinitely many of them-- 01:09:17.170 --> 01:09:20.050 align:middle line:84% or that's not quite the same-- but all but finitely many 01:09:20.050 --> 01:09:24.850 align:middle line:84% of them must exist in any ball of radius epsilon, right? 01:09:24.850 --> 01:09:27.590 align:middle line:84% And we just showed here that there are, in fact, 01:09:27.590 --> 01:09:32.420 align:middle line:84% infinitely many that don't exist in a ball of radius epsilon. 01:09:32.420 --> 01:09:35.950 align:middle line:84% So that proves the other direction. 01:09:35.950 --> 01:09:39.069 align:middle line:90% 01:09:39.069 --> 01:09:41.430 align:middle line:84% This is known as sequential continuity, sometimes. 01:09:41.430 --> 01:09:44.970 align:middle line:84% If you want to study this in more generality, as opposed 01:09:44.970 --> 01:09:48.630 align:middle line:84% to assuming continuous, you can assume this definition 01:09:48.630 --> 01:09:50.560 align:middle line:90% on a topological space. 01:09:50.560 --> 01:09:52.359 align:middle line:90% And that's slightly different. 01:09:52.359 --> 01:09:54.180 align:middle line:84% But for metric spaces, the two definitions 01:09:54.180 --> 01:09:57.210 align:middle line:84% are the same, which will be the case a lot of the time. 01:09:57.210 --> 01:09:59.880 align:middle line:84% A lot of the time in this class, a definition that's 01:09:59.880 --> 01:10:01.770 align:middle line:84% specific to metric spaces will imply 01:10:01.770 --> 01:10:04.860 align:middle line:84% other very useful definitions that might not 01:10:04.860 --> 01:10:06.960 align:middle line:90% hold in more generality. 01:10:06.960 --> 01:10:08.825 align:middle line:90% OK. 01:10:08.825 --> 01:10:10.450 align:middle line:84% I just have one more statement to show, 01:10:10.450 --> 01:10:15.420 align:middle line:84% which is how continuity relates to open sets. 01:10:15.420 --> 01:10:18.570 align:middle line:84% Let me write out what this lemma will be. 01:10:18.570 --> 01:10:20.610 align:middle line:90% Oh, sorry. 01:10:20.610 --> 01:10:22.110 align:middle line:90% This is poor board manners. 01:10:22.110 --> 01:10:27.510 align:middle line:84% I should have raised this and lowered this. 01:10:27.510 --> 01:10:30.190 align:middle line:90% OK, OK. 01:10:30.190 --> 01:10:33.197 align:middle line:90% 01:10:33.197 --> 01:10:34.030 align:middle line:90% Just one more lemma. 01:10:34.030 --> 01:10:40.390 align:middle line:90% 01:10:40.390 --> 01:10:47.950 align:middle line:84% f is continuous at c if and only if-- 01:10:47.950 --> 01:10:50.020 align:middle line:84% oh, I need to state a definition first. 01:10:50.020 --> 01:10:52.030 align:middle line:90% I'll state it over here. 01:10:52.030 --> 01:11:10.360 align:middle line:84% Definition-- a neighborhood of a point y is simply an open set 01:11:10.360 --> 01:11:11.230 align:middle line:90% that contains y-- 01:11:11.230 --> 01:11:16.240 align:middle line:90% 01:11:16.240 --> 01:11:17.200 align:middle line:90% containing y. 01:11:17.200 --> 01:11:21.027 align:middle line:90% 01:11:21.027 --> 01:11:22.610 align:middle line:84% For our purposes, though, you can just 01:11:22.610 --> 01:11:24.652 align:middle line:84% think of this open neighborhood as a ball, right? 01:11:24.652 --> 01:11:27.020 align:middle line:84% Because I've stated that every open set 01:11:27.020 --> 01:11:31.230 align:middle line:84% can be written as a union of infinitely many balls. 01:11:31.230 --> 01:11:34.065 align:middle line:84% So this is what a definition of a neighborhood is, 01:11:34.065 --> 01:11:35.690 align:middle line:84% and that lets us state the lemma, which 01:11:35.690 --> 01:11:49.670 align:middle line:84% is for every open neighborhood of f of c-- 01:11:49.670 --> 01:11:50.585 align:middle line:90% this will be our y-- 01:11:50.585 --> 01:11:54.080 align:middle line:90% 01:11:54.080 --> 01:11:59.030 align:middle line:84% then we have that the inverse image, f inverse of-- 01:11:59.030 --> 01:11:59.530 align:middle line:90% oh, sorry. 01:11:59.530 --> 01:12:01.420 align:middle line:84% I should say every open neighborhood 01:12:01.420 --> 01:12:12.655 align:middle line:84% U of f of c, f inverse of U is an open neighborhood of f 01:12:12.655 --> 01:12:16.370 align:middle line:84% of c-- oh, sorry, of c because we're in the inverse image. 01:12:16.370 --> 01:12:17.050 align:middle line:90% OK. 01:12:17.050 --> 01:12:21.890 align:middle line:84% Has anyone seen this definition before regarding open sets? 01:12:21.890 --> 01:12:23.780 align:middle line:90% Yeah. 01:12:23.780 --> 01:12:26.030 align:middle line:84% It doesn't come up all the time because it 01:12:26.030 --> 01:12:27.145 align:middle line:90% is essentially topology. 01:12:27.145 --> 01:12:28.520 align:middle line:84% If you take a topology class, you 01:12:28.520 --> 01:12:30.470 align:middle line:90% would see this definition of-- 01:12:30.470 --> 01:12:32.720 align:middle line:84% this would be the definition of continuity 01:12:32.720 --> 01:12:35.430 align:middle line:90% in a topological class. 01:12:35.430 --> 01:12:38.000 align:middle line:84% But here we're going to actually prove it for a metric space. 01:12:38.000 --> 01:12:40.042 align:middle line:84% And because not all of y'all have seen it before, 01:12:40.042 --> 01:12:41.690 align:middle line:84% we'll just go through that proof. 01:12:41.690 --> 01:12:48.880 align:middle line:84% So proof-- we'll prove the downwards direction. 01:12:48.880 --> 01:12:54.780 align:middle line:84% So we have that f is continuous at c. 01:12:54.780 --> 01:13:09.920 align:middle line:84% And I'm going to let U be an open neighborhood of f of c. 01:13:09.920 --> 01:13:13.670 align:middle line:84% And so then I want to show that f inverse of c-- 01:13:13.670 --> 01:13:15.320 align:middle line:90% oh, sorry, f inverse of U-- 01:13:15.320 --> 01:13:19.730 align:middle line:84% is an open set of c, open neighborhood of c, right? 01:13:19.730 --> 01:13:22.532 align:middle line:90% What I'm going to do instead-- 01:13:22.532 --> 01:13:24.740 align:middle line:84% instead of considering this entire open neighborhood, 01:13:24.740 --> 01:13:29.780 align:middle line:84% I'm going to consider a ball of radius epsilon around f of c. 01:13:29.780 --> 01:13:34.280 align:middle line:84% So because U is open, we know that there exists an epsilon 01:13:34.280 --> 01:13:38.180 align:middle line:84% bigger than 0 such that the ball of radius 01:13:38.180 --> 01:13:47.930 align:middle line:84% epsilon with metric dy around f of c is contained in U, right? 01:13:47.930 --> 01:13:50.900 align:middle line:84% We can find a ball around f of c such that is contained in U 01:13:50.900 --> 01:13:52.760 align:middle line:90% because it's open. 01:13:52.760 --> 01:13:53.760 align:middle line:90% What does this tell you? 01:13:53.760 --> 01:13:59.280 align:middle line:84% Well, by continuity, furthermore, there 01:13:59.280 --> 01:14:01.980 align:middle line:84% exists a delta bigger than 0 such 01:14:01.980 --> 01:14:14.370 align:middle line:84% that if the distance from x to c is less than delta, 01:14:14.370 --> 01:14:21.450 align:middle line:84% then the distance of f of x to f of c is less than epsilon. 01:14:21.450 --> 01:14:23.290 align:middle line:90% What does this tell you? 01:14:23.290 --> 01:14:26.070 align:middle line:84% Well, this tells you that the image 01:14:26.070 --> 01:14:31.200 align:middle line:84% of the ball of radius delta around c 01:14:31.200 --> 01:14:39.190 align:middle line:84% is contained in the ball of radius epsilon around f of c, 01:14:39.190 --> 01:14:39.690 align:middle line:90% right? 01:14:39.690 --> 01:14:42.340 align:middle line:84% Because if I look at the images of these points, 01:14:42.340 --> 01:14:44.700 align:middle line:84% I'm going to get that the distance is less than epsilon 01:14:44.700 --> 01:14:46.490 align:middle line:90% around f of c. 01:14:46.490 --> 01:14:47.490 align:middle line:90% And what you can check-- 01:14:47.490 --> 01:14:50.400 align:middle line:84% I won't state this right now, or I won't prove it-- 01:14:50.400 --> 01:14:53.130 align:middle line:84% but you can check that if I do f inverse here, 01:14:53.130 --> 01:14:56.460 align:middle line:84% it's going to be contained in f inverse here. 01:14:56.460 --> 01:14:57.810 align:middle line:90% Why is this helpful? 01:14:57.810 --> 01:15:02.610 align:middle line:84% Because this is a subset of f inverse of U. 01:15:02.610 --> 01:15:06.115 align:middle line:84% So what we've done is we've gone from an open neighborhood of f 01:15:06.115 --> 01:15:09.990 align:middle line:84% of c to an open neighborhood of c, which is 01:15:09.990 --> 01:15:11.910 align:middle line:90% what we wanted to show, right? 01:15:11.910 --> 01:15:15.020 align:middle line:84% We wanted to show that f inverse of U 01:15:15.020 --> 01:15:17.682 align:middle line:90% is an open neighborhood of c. 01:15:17.682 --> 01:15:19.570 align:middle line:90% OK. 01:15:19.570 --> 01:15:21.905 align:middle line:84% So we now need to prove the other direction, which is 01:15:21.905 --> 01:15:23.155 align:middle line:90% sometimes a little bit harder. 01:15:23.155 --> 01:15:31.500 align:middle line:90% 01:15:31.500 --> 01:15:34.710 align:middle line:84% Only a little bit harder, but should 01:15:34.710 --> 01:15:35.820 align:middle line:90% be able to fit right here. 01:15:35.820 --> 01:15:40.266 align:middle line:90% 01:15:40.266 --> 01:15:45.640 align:middle line:84% OK, so now we're going to suppose that let epsilon 01:15:45.640 --> 01:15:47.980 align:middle line:90% be bigger than 0. 01:15:47.980 --> 01:15:50.740 align:middle line:84% And consider the open neighborhood of f of c 01:15:50.740 --> 01:15:54.080 align:middle line:90% given by the ball-- 01:15:54.080 --> 01:16:01.640 align:middle line:84% and consider the ball of radius epsilon around f of c. 01:16:01.640 --> 01:16:04.240 align:middle line:90% 01:16:04.240 --> 01:16:07.570 align:middle line:84% Then we know, by assumption, that 01:16:07.570 --> 01:16:13.510 align:middle line:84% f inverse of this ball of radius epsilon of f of c-- 01:16:13.510 --> 01:16:22.390 align:middle line:84% 1, 2, 3-- is an open neighborhood of c. 01:16:22.390 --> 01:16:25.630 align:middle line:84% But because this is an open set around c, 01:16:25.630 --> 01:16:29.560 align:middle line:84% there exists a delta bigger than 0 such 01:16:29.560 --> 01:16:35.150 align:middle line:84% that the ball of radius delta around c 01:16:35.150 --> 01:16:39.560 align:middle line:84% is contained in f inverse of the ball of radius epsilon of f 01:16:39.560 --> 01:16:42.883 align:middle line:90% of c, right? 01:16:42.883 --> 01:16:44.050 align:middle line:90% Because this is an open set. 01:16:44.050 --> 01:16:47.290 align:middle line:84% And therefore, we know that there exists a radius such 01:16:47.290 --> 01:16:49.900 align:middle line:84% that the ball of radius delta of that radius 01:16:49.900 --> 01:16:51.655 align:middle line:90% is contained in the open set. 01:16:51.655 --> 01:16:52.780 align:middle line:90% And then we're done, right? 01:16:52.780 --> 01:16:54.670 align:middle line:84% Because then we can apply f to both sides. 01:16:54.670 --> 01:16:57.130 align:middle line:84% And we get that f is a ball of radius delta around 01:16:57.130 --> 01:17:01.577 align:middle line:84% c is contained in the ball of radius epsilon of f 01:17:01.577 --> 01:17:05.350 align:middle line:84% of c, which is exactly the statement of continuity. 01:17:05.350 --> 01:17:08.290 align:middle line:84% You can just write this out in terms of the metrics, right? 01:17:08.290 --> 01:17:13.210 align:middle line:84% Because then the distance from x to c being less than delta 01:17:13.210 --> 01:17:18.460 align:middle line:84% implies that the distance from f of x to f of c 01:17:18.460 --> 01:17:19.640 align:middle line:90% is less than epsilon. 01:17:19.640 --> 01:17:21.590 align:middle line:90% So this concludes the proof. 01:17:21.590 --> 01:17:22.570 align:middle line:90% OK. 01:17:22.570 --> 01:17:25.270 align:middle line:84% This was the general theory of metric spaces. 01:17:25.270 --> 01:17:27.070 align:middle line:84% We've talked about convergent sequences, 01:17:27.070 --> 01:17:29.500 align:middle line:84% which was essentially just like the real numbers. 01:17:29.500 --> 01:17:32.110 align:middle line:84% We talked about open sets-- specifically, 01:17:32.110 --> 01:17:34.550 align:middle line:84% the topological properties of them, 01:17:34.550 --> 01:17:36.410 align:middle line:90% which is important for topology. 01:17:36.410 --> 01:17:41.680 align:middle line:84% And then we related those two concepts to continuity, right? 01:17:41.680 --> 01:17:43.870 align:middle line:84% We talked about how continuity is sequentially 01:17:43.870 --> 01:17:47.650 align:middle line:84% continuous at a point c if it's continuous on the metric space. 01:17:47.650 --> 01:17:50.020 align:middle line:84% And we've talked about how continuity can be defined 01:17:50.020 --> 01:17:51.610 align:middle line:90% in terms of these open sets. 01:17:51.610 --> 01:17:54.160 align:middle line:84% This is by no means a simple connection. 01:17:54.160 --> 01:17:56.810 align:middle line:84% It can take a while to feel comfortable with these ideas, 01:17:56.810 --> 01:18:00.160 align:middle line:84% but I highly suggest, if you're finding this a little bit 01:18:00.160 --> 01:18:02.800 align:middle line:84% confusing, to try and draw out the pictures. 01:18:02.800 --> 01:18:05.590 align:middle line:90% Why does this conclusion hold? 01:18:05.590 --> 01:18:08.180 align:middle line:84% Why can I apply f to both sides and have things to be fine? 01:18:08.180 --> 01:18:09.970 align:middle line:84% It's just a practice in set theory, 01:18:09.970 --> 01:18:12.660 align:middle line:84% but it is helpful to do at some point. 01:18:12.660 --> 01:18:13.540 align:middle line:90% OK? 01:18:13.540 --> 01:18:14.770 align:middle line:84% Starting next time, we're going to start 01:18:14.770 --> 01:18:16.312 align:middle line:84% talking about something you might not 01:18:16.312 --> 01:18:18.550 align:middle line:84% have seen before at all known as compact sets. 01:18:18.550 --> 01:18:20.530 align:middle line:90% Until then have a great weekend. 01:18:20.530 --> 01:18:24.810 align:middle line:84% Remember that the p set is due on the 13th, OK? 01:18:24.810 --> 01:18:27.890 align:middle line:90% All right, have a great day. 01:18:27.890 --> 01:18:34.000 align:middle line:90%