WEBVTT 00:00:00.000 --> 00:00:01.968 align:middle line:90% [SQUEAKING] 00:00:01.968 --> 00:00:03.936 align:middle line:90% [RUSTLING] 00:00:03.936 --> 00:00:07.380 align:middle line:90% [CLICKING] 00:00:07.380 --> 00:00:15.440 align:middle line:90% 00:00:15.440 --> 00:00:19.790 align:middle line:84% PROFESSOR: But let me just quickly review 00:00:19.790 --> 00:00:22.950 align:middle line:84% some what we have done so far in the class. 00:00:22.950 --> 00:00:32.304 align:middle line:84% So first, we talked about R as an ordered set and fields. 00:00:32.304 --> 00:00:36.470 align:middle line:90% 00:00:36.470 --> 00:00:39.805 align:middle line:84% Then we have talked about sequences and limits. 00:00:39.805 --> 00:00:49.730 align:middle line:90% 00:00:49.730 --> 00:00:51.523 align:middle line:90% And then we talked about series. 00:00:51.523 --> 00:00:55.370 align:middle line:90% 00:00:55.370 --> 00:00:58.130 align:middle line:84% And then, finally, we have talked about-- 00:00:58.130 --> 00:01:00.720 align:middle line:84% and this is where we will continue today-- 00:01:00.720 --> 00:01:02.960 align:middle line:90% we talked about metric spaces. 00:01:02.960 --> 00:01:06.810 align:middle line:90% 00:01:06.810 --> 00:01:13.770 align:middle line:84% And so the idea here is that we want to-- 00:01:13.770 --> 00:01:20.760 align:middle line:84% there's a lot of things from R that actually generalize 00:01:20.760 --> 00:01:23.370 align:middle line:90% and make sense. 00:01:23.370 --> 00:01:26.500 align:middle line:84% And some of those are sequences and limits. 00:01:26.500 --> 00:01:30.160 align:middle line:84% So they generalize from R to metric spaces. 00:01:30.160 --> 00:01:32.760 align:middle line:84% There are other things that don't generalize. 00:01:32.760 --> 00:01:33.940 align:middle line:90% It's not an order. 00:01:33.940 --> 00:01:37.650 align:middle line:84% A general metric space is not-- it doesn't necessarily 00:01:37.650 --> 00:01:39.550 align:middle line:90% come with an ordering. 00:01:39.550 --> 00:01:42.540 align:middle line:90% And it's not like a field. 00:01:42.540 --> 00:01:48.190 align:middle line:84% And in particular, typically, you can't sum elements. 00:01:48.190 --> 00:01:51.490 align:middle line:84% So a series doesn't make sense either. 00:01:51.490 --> 00:01:56.700 align:middle line:84% But these things here make sense in a metric space. 00:01:56.700 --> 00:02:02.540 align:middle line:84% And so let me just remind you what is a metric space. 00:02:02.540 --> 00:02:12.270 align:middle line:90% So a metric space is a set. 00:02:12.270 --> 00:02:14.750 align:middle line:90% So x is a set. 00:02:14.750 --> 00:02:17.910 align:middle line:84% And then there is a distance function. 00:02:17.910 --> 00:02:24.680 align:middle line:84% So this is a map from x cross x into r. 00:02:24.680 --> 00:02:32.240 align:middle line:84% And for each two pair of elements x and y, 00:02:32.240 --> 00:02:35.760 align:middle line:90% in x you have that-- 00:02:35.760 --> 00:02:40.770 align:middle line:84% let me make it a capital X, like this. 00:02:40.770 --> 00:02:46.430 align:middle line:84% For every two elements in this set, the distance between them 00:02:46.430 --> 00:02:58.710 align:middle line:84% should be positive and it should be nonnegative and equal to 0. 00:02:58.710 --> 00:03:00.570 align:middle line:84% The distance between two points should 00:03:00.570 --> 00:03:06.240 align:middle line:84% be 0 if and only if these points are the same. 00:03:06.240 --> 00:03:10.030 align:middle line:90% And then it should be symmetric. 00:03:10.030 --> 00:03:13.540 align:middle line:84% So property 2 is that it should be symmetric. 00:03:13.540 --> 00:03:18.870 align:middle line:84% So it doesn't matter whether you go from x to y or from y to x. 00:03:18.870 --> 00:03:20.360 align:middle line:90% The distance is the same. 00:03:20.360 --> 00:03:23.520 align:middle line:90% 00:03:23.520 --> 00:03:28.560 align:middle line:84% And the third, which is the most nontrivial part, 00:03:28.560 --> 00:03:31.180 align:middle line:90% is the triangle inequality. 00:03:31.180 --> 00:03:32.676 align:middle line:90% So this here is symmetric. 00:03:32.676 --> 00:03:36.960 align:middle line:90% 00:03:36.960 --> 00:03:40.080 align:middle line:84% And the third property is that if you're 00:03:40.080 --> 00:03:45.000 align:middle line:84% looking at x, y, and z, so you have three points, 00:03:45.000 --> 00:03:55.720 align:middle line:84% and you go from x to C, then it's certainly not further than 00:03:55.720 --> 00:04:01.820 align:middle line:84% if you first go to from x to y and then go from y to z. 00:04:01.820 --> 00:04:05.590 align:middle line:84% So this should be less than or equal to the distance from x 00:04:05.590 --> 00:04:09.760 align:middle line:84% to y plus the distance from y to z. 00:04:09.760 --> 00:04:14.080 align:middle line:84% And so this third property here is what's 00:04:14.080 --> 00:04:16.214 align:middle line:90% called the triangle inequality. 00:04:16.214 --> 00:04:19.329 align:middle line:90% 00:04:19.329 --> 00:04:21.691 align:middle line:84% So this is the triangle inequality. 00:04:21.691 --> 00:04:29.560 align:middle line:90% 00:04:29.560 --> 00:04:34.760 align:middle line:84% Now so that's what a metric space is. 00:04:34.760 --> 00:04:44.680 align:middle line:84% And again, so the canonical example of a metric space 00:04:44.680 --> 00:04:49.310 align:middle line:84% is where you're taking x equal to the real numbers. 00:04:49.310 --> 00:04:52.630 align:middle line:84% And then for two true real numbers, 00:04:52.630 --> 00:04:54.660 align:middle line:84% the distance between them is just where 00:04:54.660 --> 00:04:58.950 align:middle line:84% you're looking at the norm of the difference. 00:04:58.950 --> 00:05:02.010 align:middle line:90% So this here is a nice distance. 00:05:02.010 --> 00:05:03.910 align:middle line:90% It's clearly nonnegative. 00:05:03.910 --> 00:05:08.530 align:middle line:84% It's clearly equal to 0 if and only if x is equal to y. 00:05:08.530 --> 00:05:10.240 align:middle line:90% It's clearly symmetric. 00:05:10.240 --> 00:05:13.020 align:middle line:84% And the triangle inequality is this inequality 00:05:13.020 --> 00:05:14.655 align:middle line:90% that we have used all the time. 00:05:14.655 --> 00:05:18.010 align:middle line:90% 00:05:18.010 --> 00:05:23.790 align:middle line:84% So this here is slightly more fancy version of the same 00:05:23.790 --> 00:05:28.530 align:middle line:90% is where you're looking at R2. 00:05:28.530 --> 00:05:37.500 align:middle line:84% So the x here is R2 Or it could even be a general n. 00:05:37.500 --> 00:05:44.160 align:middle line:84% And then if you think about two elements in Rn 00:05:44.160 --> 00:05:48.540 align:middle line:84% and you think about, say, x here as-- 00:05:48.540 --> 00:05:55.040 align:middle line:84% it's written as x1, x2, up to xn, 00:05:55.040 --> 00:05:58.540 align:middle line:84% where each of these xi's are in R, 00:05:58.540 --> 00:06:02.685 align:middle line:84% and there are n many of them, and likewise for y. 00:06:02.685 --> 00:06:09.520 align:middle line:90% 00:06:09.520 --> 00:06:14.830 align:middle line:84% And then the distance here the natural distance 00:06:14.830 --> 00:06:18.790 align:middle line:90% is where you're taking-- 00:06:18.790 --> 00:06:21.290 align:middle line:84% this is what's called the Euclidean distance. 00:06:21.290 --> 00:06:23.480 align:middle line:90% It's given by Pythagoras. 00:06:23.480 --> 00:06:29.420 align:middle line:84% And so it's the square root of the sum of the squares, 00:06:29.420 --> 00:06:39.190 align:middle line:84% so x1 minus y1 squared plus x2 minus y2 squared plus 00:06:39.190 --> 00:06:46.200 align:middle line:84% and then all the way up to xn minus yn squared. 00:06:46.200 --> 00:06:48.890 align:middle line:90% 00:06:48.890 --> 00:06:53.600 align:middle line:84% And so you see that if n here is equal to 1, well, 00:06:53.600 --> 00:06:59.090 align:middle line:84% then each element is just a single element in R. 00:06:59.090 --> 00:07:03.170 align:middle line:84% And so then this thing here is really the same as that 00:07:03.170 --> 00:07:07.580 align:middle line:84% because it's the square of x minus y. 00:07:07.580 --> 00:07:09.413 align:middle line:84% It's the square of this distance. 00:07:09.413 --> 00:07:10.830 align:middle line:84% And then you take the square root. 00:07:10.830 --> 00:07:13.220 align:middle line:90% So it's a norm. 00:07:13.220 --> 00:07:17.960 align:middle line:84% So both of these is called the Euclidean-- 00:07:17.960 --> 00:07:20.470 align:middle line:84% usually referred to as a Euclidean distance. 00:07:20.470 --> 00:07:26.990 align:middle line:90% 00:07:26.990 --> 00:07:32.900 align:middle line:84% So we looked at a few other examples. 00:07:32.900 --> 00:07:38.600 align:middle line:84% And I won't look at all of them again, but just two more. 00:07:38.600 --> 00:07:41.155 align:middle line:84% One is this with the box distance. 00:07:41.155 --> 00:07:45.350 align:middle line:90% 00:07:45.350 --> 00:07:50.210 align:middle line:84% And let me state it only in R2, but just like this, 00:07:50.210 --> 00:07:52.340 align:middle line:90% you could state it in any Rn. 00:07:52.340 --> 00:08:00.440 align:middle line:84% So the spaces are true, and the distance between two elements-- 00:08:00.440 --> 00:08:02.270 align:middle line:90% so again, we write it like this. 00:08:02.270 --> 00:08:05.710 align:middle line:84% And we think about x as being x1 comma x2, 00:08:05.710 --> 00:08:11.800 align:middle line:84% where these are real numbers, y equal to y1 comma y2. 00:08:11.800 --> 00:08:20.110 align:middle line:84% And then this distance here is the sum of these distances. 00:08:20.110 --> 00:08:26.590 align:middle line:84% So this is a plus x2 minus y2, like that. 00:08:26.590 --> 00:08:29.210 align:middle line:84% And this is clearly, again, symmetric. 00:08:29.210 --> 00:08:30.800 align:middle line:90% It's clearly nonnegative. 00:08:30.800 --> 00:08:36.679 align:middle line:84% It's clearly equal to 0 if and only if x and y are the same. 00:08:36.679 --> 00:08:39.280 align:middle line:84% And the triangle inequality-- well, that just 00:08:39.280 --> 00:08:43.360 align:middle line:84% follows quite easily from the triangle inequality 00:08:43.360 --> 00:08:46.730 align:middle line:90% for this distance. 00:08:46.730 --> 00:08:49.440 align:middle line:90% Then there is one more example. 00:08:49.440 --> 00:08:51.800 align:middle line:90% I mean, there's other-- 00:08:51.800 --> 00:08:54.410 align:middle line:84% but there's one more example that is often 00:08:54.410 --> 00:08:56.390 align:middle line:90% useful to keep in mind. 00:08:56.390 --> 00:09:01.670 align:middle line:84% And this is the last one I will review. 00:09:01.670 --> 00:09:05.340 align:middle line:84% And so this is the continuous function. 00:09:05.340 --> 00:09:14.690 align:middle line:84% So if you take an interval a and b are real numbers, 00:09:14.690 --> 00:09:16.490 align:middle line:84% and you're looking at the interval 00:09:16.490 --> 00:09:20.490 align:middle line:84% from a to b where both endpoints are included, 00:09:20.490 --> 00:09:23.660 align:middle line:84% and then you look at the continuous function 00:09:23.660 --> 00:09:26.660 align:middle line:90% on this interval. 00:09:26.660 --> 00:09:32.450 align:middle line:84% And then if you have two elements f and g-- 00:09:32.450 --> 00:09:34.140 align:middle line:90% so these are continuous. 00:09:34.140 --> 00:09:38.330 align:middle line:84% Each of them are continuous functions. 00:09:38.330 --> 00:09:43.350 align:middle line:84% Then the distance between these two continuous functions 00:09:43.350 --> 00:09:51.030 align:middle line:84% is equal to the max over x And then 00:09:51.030 --> 00:09:57.450 align:middle line:84% f of x minus g of x, right then, so x here 00:09:57.450 --> 00:09:59.711 align:middle line:90% is in the interval from a to b. 00:09:59.711 --> 00:10:03.990 align:middle line:90% 00:10:03.990 --> 00:10:06.640 align:middle line:90% And so this thing here-- 00:10:06.640 --> 00:10:09.780 align:middle line:84% so this here was on the midterm, this particular example. 00:10:09.780 --> 00:10:14.310 align:middle line:84% And this thing here that the max is achieved 00:10:14.310 --> 00:10:17.640 align:middle line:84% is just like if you're looking at these two functions, then 00:10:17.640 --> 00:10:20.890 align:middle line:84% this is the difference between the two functions. 00:10:20.890 --> 00:10:23.170 align:middle line:90% So this is also continuous. 00:10:23.170 --> 00:10:26.400 align:middle line:84% And this means that the maximum at the minimum 00:10:26.400 --> 00:10:28.890 align:middle line:84% by the extreme value theorem is achieved 00:10:28.890 --> 00:10:30.810 align:middle line:90% on an interval of this form. 00:10:30.810 --> 00:10:34.470 align:middle line:84% So this thing here is really achieved, 00:10:34.470 --> 00:10:38.050 align:middle line:90% and it's clearly nonnegative. 00:10:38.050 --> 00:10:41.750 align:middle line:84% And it's easy to see that it has the other property 00:10:41.750 --> 00:10:43.460 align:middle line:90% of a metric space. 00:10:43.460 --> 00:10:47.300 align:middle line:84% So these are just the examples that we talked about. 00:10:47.300 --> 00:10:51.260 align:middle line:90% And I just want to remind you. 00:10:51.260 --> 00:10:58.970 align:middle line:84% And so then the question is what generalizes from-- 00:10:58.970 --> 00:11:03.870 align:middle line:84% so we started with a class talking about R. 00:11:03.870 --> 00:11:07.290 align:middle line:90% And so it was an order field. 00:11:07.290 --> 00:11:08.870 align:middle line:84% But a lot of the things we talked 00:11:08.870 --> 00:11:11.180 align:middle line:84% about for R, the subsequent things 00:11:11.180 --> 00:11:16.340 align:middle line:84% we talked about, sequences and limits, they, as we will see, 00:11:16.340 --> 00:11:21.110 align:middle line:84% make total sense in a general metric space 00:11:21.110 --> 00:11:26.420 align:middle line:84% and has important consequences, including 00:11:26.420 --> 00:11:31.700 align:middle line:84% to solve differential equations and solving 00:11:31.700 --> 00:11:33.380 align:middle line:84% all kinds of equations, including 00:11:33.380 --> 00:11:35.900 align:middle line:90% differential equations. 00:11:35.900 --> 00:11:44.190 align:middle line:84% So now let me remind you also, so we talked about that 00:11:44.190 --> 00:11:54.575 align:middle line:84% if this is a metric space, then a sequence in a metric space. 00:11:54.575 --> 00:11:59.550 align:middle line:90% 00:11:59.550 --> 00:12:01.852 align:middle line:90% This is just a map. 00:12:01.852 --> 00:12:10.170 align:middle line:84% This is a map from the natural numbers into x. 00:12:10.170 --> 00:12:14.190 align:middle line:84% So a sequence is like a map like this. 00:12:14.190 --> 00:12:17.790 align:middle line:84% And so usually, the image of the n 00:12:17.790 --> 00:12:22.800 align:middle line:84% element, this we will typically denote, again, like this, xn. 00:12:22.800 --> 00:12:25.740 align:middle line:90% So this is what a sequence is. 00:12:25.740 --> 00:12:35.610 align:middle line:84% A subsequence-- this is defined very similar to what 00:12:35.610 --> 00:12:37.880 align:middle line:90% we did on the real line. 00:12:37.880 --> 00:12:41.680 align:middle line:84% So a subsequence is the composition 00:12:41.680 --> 00:13:06.700 align:middle line:84% of f with a strictly increasing map g from the natural numbers 00:13:06.700 --> 00:13:08.620 align:middle line:90% into n. 00:13:08.620 --> 00:13:13.600 align:middle line:84% And then the sequence is then where 00:13:13.600 --> 00:13:16.660 align:middle line:84% you're looking where you apply first g, 00:13:16.660 --> 00:13:20.090 align:middle line:90% and then you apply the f. 00:13:20.090 --> 00:13:24.340 align:middle line:84% So when you first apply g, then it 00:13:24.340 --> 00:13:26.270 align:middle line:84% gives you another natural number. 00:13:26.270 --> 00:13:28.600 align:middle line:84% And so this thing here, the image of this 00:13:28.600 --> 00:13:30.760 align:middle line:90% is then a element in x. 00:13:30.760 --> 00:13:32.650 align:middle line:90% So that's a subsequence. 00:13:32.650 --> 00:13:35.150 align:middle line:84% And again, this was strictly increasing 00:13:35.150 --> 00:13:39.750 align:middle line:84% guarantee you that the elements appear in the same order. 00:13:39.750 --> 00:13:41.580 align:middle line:84% You're picking a selection of them, 00:13:41.580 --> 00:13:43.160 align:middle line:84% but they appear in the same order 00:13:43.160 --> 00:13:44.770 align:middle line:90% as in the original sequence. 00:13:44.770 --> 00:13:48.230 align:middle line:90% 00:13:48.230 --> 00:13:51.670 align:middle line:84% So sequences and subsequences make sense. 00:13:51.670 --> 00:13:54.530 align:middle line:90% 00:13:54.530 --> 00:13:56.667 align:middle line:90% We can then also talk about-- 00:13:56.667 --> 00:13:58.250 align:middle line:84% and I know we talked about, but I just 00:13:58.250 --> 00:14:00.200 align:middle line:84% want to-- because of the break, I just 00:14:00.200 --> 00:14:05.120 align:middle line:84% want to review quickly a few of the things we talked 00:14:05.120 --> 00:14:07.610 align:middle line:90% about right before the break. 00:14:07.610 --> 00:14:22.390 align:middle line:84% So one is that a convergent sequence in a metric space-- 00:14:22.390 --> 00:14:26.810 align:middle line:90% 00:14:26.810 --> 00:14:33.640 align:middle line:84% so if this here is a metric space and xn 00:14:33.640 --> 00:14:49.390 align:middle line:84% is a sequence in xd, then we say that we 00:14:49.390 --> 00:15:02.595 align:middle line:84% say that xn converges to x if the following hold. 00:15:02.595 --> 00:15:10.870 align:middle line:90% 00:15:10.870 --> 00:15:21.310 align:middle line:84% That for all epsilon greater than 0, 00:15:21.310 --> 00:15:33.320 align:middle line:84% there exists capital N such that if little n is 00:15:33.320 --> 00:15:40.340 align:middle line:84% equal to capital N, then xn the distance between xn 00:15:40.340 --> 00:15:45.440 align:middle line:84% and this limit should be less than epsilon. 00:15:45.440 --> 00:15:49.790 align:middle line:84% So that's exact parallel to how we defined 00:15:49.790 --> 00:15:53.660 align:middle line:84% the limit of a sequence or that a sequence is converging 00:15:53.660 --> 00:15:56.930 align:middle line:90% if it was a sequence in R. 00:15:56.930 --> 00:16:03.320 align:middle line:84% And likewise, we can define a sequence in a metric space. 00:16:03.320 --> 00:16:22.670 align:middle line:84% So we define a Cauchy sequence in a metric space x, d. 00:16:22.670 --> 00:16:26.460 align:middle line:84% So this is a sequence so that for all-- 00:16:26.460 --> 00:16:35.460 align:middle line:84% so this is a sequence xn in such that 00:16:35.460 --> 00:16:41.040 align:middle line:84% for all epsilon greater than 0, there 00:16:41.040 --> 00:16:54.930 align:middle line:84% exists a capital N such that if little m and little n 00:16:54.930 --> 00:16:58.390 align:middle line:84% are bigger or equal to capital N. 00:16:58.390 --> 00:17:00.210 align:middle line:84% So remember that a Cauchy sequence 00:17:00.210 --> 00:17:03.190 align:middle line:84% means that the tail is bunching together. 00:17:03.190 --> 00:17:06.240 align:middle line:84% So this means that if you are sufficiently far out, 00:17:06.240 --> 00:17:13.770 align:middle line:84% then these elements, xn and xm, are epsilon close together. 00:17:13.770 --> 00:17:17.520 align:middle line:84% So this is what it means for a sequence 00:17:17.520 --> 00:17:20.609 align:middle line:90% to be a Cauchy sequence. 00:17:20.609 --> 00:17:25.050 align:middle line:84% And so now we have this theorem, and I 00:17:25.050 --> 00:17:27.670 align:middle line:84% think I mentioned it before the break. 00:17:27.670 --> 00:17:29.780 align:middle line:84% But let's just quickly run through it 00:17:29.780 --> 00:17:31.940 align:middle line:90% because it's very short. 00:17:31.940 --> 00:17:39.530 align:middle line:84% So if you have a-- so if this is a metric space, 00:17:39.530 --> 00:17:57.830 align:middle line:84% then the theorem is that any convergent sequence is also 00:17:57.830 --> 00:17:58.810 align:middle line:90% a Cauchy sequence. 00:17:58.810 --> 00:18:06.140 align:middle line:90% 00:18:06.140 --> 00:18:08.005 align:middle line:90% And the proof is like this. 00:18:08.005 --> 00:18:10.850 align:middle line:90% 00:18:10.850 --> 00:18:25.680 align:middle line:84% So given epsilon, since xn is a convergent sequence, 00:18:25.680 --> 00:18:28.680 align:middle line:84% so you assume that x is a convergence sequence-- let's 00:18:28.680 --> 00:18:30.100 align:middle line:90% call the limit x. 00:18:30.100 --> 00:18:35.430 align:middle line:84% Since xn converges to x, this means 00:18:35.430 --> 00:18:49.590 align:middle line:84% that there exists capital N such that if little n is bigger 00:18:49.590 --> 00:18:53.610 align:middle line:84% or equal to capital N, then the distance between little 00:18:53.610 --> 00:19:03.390 align:middle line:84% x, little n, and nx here, this distance here is less than-- 00:19:03.390 --> 00:19:07.200 align:middle line:90% and then I use epsilon over 2. 00:19:07.200 --> 00:19:15.570 align:middle line:84% And so this means that, therefore, 00:19:15.570 --> 00:19:23.220 align:middle line:84% if little m and little n are both bigger than capital N then 00:19:23.220 --> 00:19:26.470 align:middle line:90% the distance between the two-- 00:19:26.470 --> 00:19:30.610 align:middle line:84% distance between these two elements here, 00:19:30.610 --> 00:19:33.790 align:middle line:84% well, by the triangle inequality, 00:19:33.790 --> 00:19:36.830 align:middle line:84% we have that this distance is less than 00:19:36.830 --> 00:19:42.820 align:middle line:84% if you go over the point x, so if you go from xn to x and then 00:19:42.820 --> 00:19:46.480 align:middle line:90% from x to xm like that. 00:19:46.480 --> 00:19:54.400 align:middle line:84% But each of these, because both of these, both little m 00:19:54.400 --> 00:19:57.260 align:middle line:84% and little n are bigger than capital N, 00:19:57.260 --> 00:20:03.010 align:middle line:84% then both of these distances are bounded by epsilon over 2. 00:20:03.010 --> 00:20:07.340 align:middle line:84% And so you see that this thing here is smaller than epsilon. 00:20:07.340 --> 00:20:10.120 align:middle line:84% So this proved that in a metric space, 00:20:10.120 --> 00:20:14.470 align:middle line:84% any convergent sequence is also a Cauchy sequence. 00:20:14.470 --> 00:20:21.670 align:middle line:84% And so we know that if the metric space is 00:20:21.670 --> 00:20:26.550 align:middle line:84% R with the usual Euclidean distance, 00:20:26.550 --> 00:20:30.300 align:middle line:84% then actually, you also-- and that 00:20:30.300 --> 00:20:33.005 align:middle line:84% was the convergence theorem, so the Cauchy. 00:20:33.005 --> 00:20:35.640 align:middle line:90% 00:20:35.640 --> 00:20:36.755 align:middle line:90% convergence theorem. 00:20:36.755 --> 00:20:41.310 align:middle line:90% 00:20:41.310 --> 00:20:51.840 align:middle line:84% So Cauchy convergence theorem-- it said that if x comma d is R 00:20:51.840 --> 00:20:56.580 align:middle line:84% with the Euclidean distance-- so I denote the Euclidean distance 00:20:56.580 --> 00:20:59.340 align:middle line:90% like this-- 00:20:59.340 --> 00:21:02.750 align:middle line:90% then we have the converse. 00:21:02.750 --> 00:21:11.550 align:middle line:90% 00:21:11.550 --> 00:21:14.222 align:middle line:90% So i.e. 00:21:14.222 --> 00:21:18.520 align:middle line:90% that any Cauchy sequence-- 00:21:18.520 --> 00:21:24.130 align:middle line:90% 00:21:24.130 --> 00:21:27.320 align:middle line:90% this is also convergent. 00:21:27.320 --> 00:21:37.258 align:middle line:90% 00:21:37.258 --> 00:21:42.410 align:middle line:84% Now we looked at a very simple example. 00:21:42.410 --> 00:21:46.790 align:middle line:84% And if I have time, I will talk about it a little bit more 00:21:46.790 --> 00:21:49.810 align:middle line:90% interesting example. 00:21:49.810 --> 00:21:54.230 align:middle line:84% Otherwise, I'll talk about that next time. 00:21:54.230 --> 00:22:00.940 align:middle line:84% So if the metric space was R with the Euclidean distance, 00:22:00.940 --> 00:22:05.570 align:middle line:84% but then we had a converse-- but if the metric space was, say-- 00:22:05.570 --> 00:22:08.510 align:middle line:84% so we looked at where the metric space was-- 00:22:08.510 --> 00:22:13.400 align:middle line:84% so where x here, the set was the interval from 0 to 1, 00:22:13.400 --> 00:22:18.620 align:middle line:84% and the distance here was just the usual Euclidean distance. 00:22:18.620 --> 00:22:27.290 align:middle line:84% So then if you're looking at the sequence xn, 1 over n, 00:22:27.290 --> 00:22:32.340 align:middle line:84% this one here, this is a sequence in this space. 00:22:32.340 --> 00:22:34.640 align:middle line:90% It's a Cauchy sequence. 00:22:34.640 --> 00:22:37.880 align:middle line:84% But it doesn't converge in this space 00:22:37.880 --> 00:22:42.140 align:middle line:84% because 0 is not part of the space. 00:22:42.140 --> 00:22:46.490 align:middle line:84% So this here is a Cauchy sequence. 00:22:46.490 --> 00:22:51.720 align:middle line:84% So you may say this is a rather trivial example, 00:22:51.720 --> 00:22:54.860 align:middle line:90% not very interesting. 00:22:54.860 --> 00:22:59.510 align:middle line:84% I'll show you another example a little bit later or next time 00:22:59.510 --> 00:23:04.280 align:middle line:84% that is more interesting but, in some sense, 00:23:04.280 --> 00:23:07.790 align:middle line:84% has some of the same feature, that the space is just not 00:23:07.790 --> 00:23:09.450 align:middle line:90% quite large enough. 00:23:09.450 --> 00:23:11.510 align:middle line:84% But it will seem more interesting. 00:23:11.510 --> 00:23:18.560 align:middle line:84% So this here is a Cauchy sequence that is not convergent. 00:23:18.560 --> 00:23:23.590 align:middle line:90% 00:23:23.590 --> 00:23:32.110 align:middle line:84% So again, the Cauchy [INAUDIBLE], so this 00:23:32.110 --> 00:23:34.030 align:middle line:90% leads to the following. 00:23:34.030 --> 00:23:36.570 align:middle line:84% This leads to the following notion here. 00:23:36.570 --> 00:23:45.520 align:middle line:90% 00:23:45.520 --> 00:23:47.530 align:middle line:84% So this is the following definition 00:23:47.530 --> 00:23:52.923 align:middle line:90% of being Cauchy complete. 00:23:52.923 --> 00:23:57.400 align:middle line:90% 00:23:57.400 --> 00:24:01.090 align:middle line:84% So a space or metric space x comma d 00:24:01.090 --> 00:24:22.876 align:middle line:84% is Cauchy complete if every Cauchy sequence is convergent. 00:24:22.876 --> 00:24:33.630 align:middle line:90% 00:24:33.630 --> 00:24:38.700 align:middle line:90% So now let me define some-- 00:24:38.700 --> 00:24:44.930 align:middle line:84% so if we begin with extending some notions from our-- 00:24:44.930 --> 00:24:47.600 align:middle line:84% so the first one I want to extend 00:24:47.600 --> 00:24:52.150 align:middle line:84% is what it means in a metric space, what is a form? 00:24:52.150 --> 00:24:59.600 align:middle line:90% 00:24:59.600 --> 00:25:04.040 align:middle line:90% So we have here-- 00:25:04.040 --> 00:25:05.650 align:middle line:90% so a metric form. 00:25:05.650 --> 00:25:10.160 align:middle line:90% 00:25:10.160 --> 00:25:13.590 align:middle line:90% So we have a metric space. 00:25:13.590 --> 00:25:17.710 align:middle line:84% We have a point in the metric space. 00:25:17.710 --> 00:25:23.970 align:middle line:84% We have some r which is a real number, positive real number. 00:25:23.970 --> 00:25:29.700 align:middle line:84% And then we define a Br of x here. 00:25:29.700 --> 00:25:41.610 align:middle line:84% This here is the set y in x So that the distance from x to y 00:25:41.610 --> 00:25:44.160 align:middle line:84% is less than or strictly less than r. 00:25:44.160 --> 00:25:49.830 align:middle line:90% 00:25:49.830 --> 00:25:53.730 align:middle line:84% And so this here is called this here is called the metric ball. 00:25:53.730 --> 00:25:57.630 align:middle line:84% This set here is called a metric ball or just 00:25:57.630 --> 00:26:11.680 align:middle line:90% a ball with radius r and set x. 00:26:11.680 --> 00:26:18.440 align:middle line:90% 00:26:18.440 --> 00:26:25.230 align:middle line:90% Now let me just illustrate it. 00:26:25.230 --> 00:26:27.960 align:middle line:84% So a lot of the intuitions about metric spaces-- 00:26:27.960 --> 00:26:31.940 align:middle line:84% but you have to be careful because, sometimes, 00:26:31.940 --> 00:26:36.980 align:middle line:84% that intuition fools you just because metric spaces are 00:26:36.980 --> 00:26:38.880 align:middle line:90% allowed to be very general. 00:26:38.880 --> 00:26:41.600 align:middle line:84% But a lot of the intuition about metric spaces 00:26:41.600 --> 00:26:46.560 align:middle line:84% come from just thinking about things in Euclidean space. 00:26:46.560 --> 00:26:49.790 align:middle line:84% So when I'm talking about Euclidean space, 00:26:49.790 --> 00:26:53.340 align:middle line:84% I'm thinking mostly about it could be on the line. 00:26:53.340 --> 00:26:57.920 align:middle line:84% So in the case where x is r or it could be in a plane 00:26:57.920 --> 00:27:02.730 align:middle line:84% where x is r2, or it could be where x is r3. 00:27:02.730 --> 00:27:05.490 align:middle line:84% And so in either of those three cases, 00:27:05.490 --> 00:27:08.810 align:middle line:84% the distance is the distance that you're used to, 00:27:08.810 --> 00:27:12.630 align:middle line:84% which is the Euclidean distance we talked about. 00:27:12.630 --> 00:27:18.130 align:middle line:84% And so in that case, if you take a point here-- 00:27:18.130 --> 00:27:21.510 align:middle line:84% so just thinking about the example 00:27:21.510 --> 00:27:24.210 align:middle line:84% where you think about the example where 00:27:24.210 --> 00:27:28.360 align:middle line:90% x is R2, then a ball here-- 00:27:28.360 --> 00:27:31.920 align:middle line:84% well, a usual ball is really just everything 00:27:31.920 --> 00:27:35.610 align:middle line:90% that is the distance r from x. 00:27:35.610 --> 00:27:38.410 align:middle line:84% So this is kind of what I defined over here. 00:27:38.410 --> 00:27:44.440 align:middle line:84% There's another thing you could also look at some other sets. 00:27:44.440 --> 00:27:52.050 align:middle line:84% So imagine that you take a line in the plane, 00:27:52.050 --> 00:27:56.850 align:middle line:84% and you could look at-- or Maybe this could even be-- 00:27:56.850 --> 00:27:58.530 align:middle line:84% it would look more interesting if I'm 00:27:58.530 --> 00:28:02.070 align:middle line:90% thinking about a line in R3. 00:28:02.070 --> 00:28:04.920 align:middle line:84% So imagine that this is now-- this 00:28:04.920 --> 00:28:07.090 align:middle line:90% is supposed to look like in R3. 00:28:07.090 --> 00:28:08.920 align:middle line:84% And then you could look at a line, 00:28:08.920 --> 00:28:13.310 align:middle line:84% and you could look at-- instead of looking at a metric ball, 00:28:13.310 --> 00:28:16.040 align:middle line:84% you could look at a tubular neighborhood. 00:28:16.040 --> 00:28:17.810 align:middle line:90% So suppose you're looking at-- 00:28:17.810 --> 00:28:19.335 align:middle line:90% that's very similar to a ball. 00:28:19.335 --> 00:28:27.850 align:middle line:90% 00:28:27.850 --> 00:28:33.080 align:middle line:84% So a tubular neighborhood, let's say a r tubular neighborhood-- 00:28:33.080 --> 00:28:36.700 align:middle line:84% this is where have some positive number r 00:28:36.700 --> 00:28:43.150 align:middle line:84% and you're looking at everything that has distance at most 00:28:43.150 --> 00:28:46.970 align:middle line:84% or let's say strictly less than r from this line. 00:28:46.970 --> 00:28:49.180 align:middle line:84% So if you do that, that would look 00:28:49.180 --> 00:28:52.600 align:middle line:84% like a cylinder, where the boundary of the cylinder 00:28:52.600 --> 00:28:56.200 align:middle line:84% is not included because I require things to have distance 00:28:56.200 --> 00:28:59.920 align:middle line:90% strictly less than r. 00:28:59.920 --> 00:29:05.700 align:middle line:84% This here would be a r tubular neighborhood of a line. 00:29:05.700 --> 00:29:11.050 align:middle line:84% So this is just to say that of notion here of a ball, 00:29:11.050 --> 00:29:13.690 align:middle line:84% you could even have something more general. 00:29:13.690 --> 00:29:17.610 align:middle line:84% And both of these things that you know well 00:29:17.610 --> 00:29:20.430 align:middle line:84% from Euclidean space, they make sense 00:29:20.430 --> 00:29:22.545 align:middle line:90% in just a general metric space. 00:29:22.545 --> 00:29:25.140 align:middle line:84% The line maybe doesn't make sense. 00:29:25.140 --> 00:29:28.380 align:middle line:84% Sometimes, often, it does, actually. 00:29:28.380 --> 00:29:30.700 align:middle line:84% But that's something more complicated. 00:29:30.700 --> 00:29:33.430 align:middle line:84% But you could just think about it as a subset. 00:29:33.430 --> 00:29:37.560 align:middle line:84% And then [? things ?] that have distance strictly less than r 00:29:37.560 --> 00:29:38.355 align:middle line:90% to that subset. 00:29:38.355 --> 00:29:44.530 align:middle line:90% 00:29:44.530 --> 00:29:48.324 align:middle line:90% So this is what it means for-- 00:29:48.324 --> 00:29:52.620 align:middle line:84% this is what a ball is, metric ball. 00:29:52.620 --> 00:29:57.930 align:middle line:84% Then there's the notion of a subset of a metric space being 00:29:57.930 --> 00:30:01.610 align:middle line:90% bounded, so a bounded subset. 00:30:01.610 --> 00:30:06.400 align:middle line:90% 00:30:06.400 --> 00:30:09.880 align:middle line:90% So there's another definition-- 00:30:09.880 --> 00:30:20.595 align:middle line:84% bounded subset of a metric space. 00:30:20.595 --> 00:30:27.820 align:middle line:90% 00:30:27.820 --> 00:30:31.480 align:middle line:90% So we have a metric space here. 00:30:31.480 --> 00:30:35.890 align:middle line:90% And we have A as a subset. 00:30:35.890 --> 00:30:56.260 align:middle line:84% And we say that A is bounded if there 00:30:56.260 --> 00:31:02.585 align:middle line:84% exists x, an element in this metric space, and d-- 00:31:02.585 --> 00:31:05.400 align:middle line:90% 00:31:05.400 --> 00:31:14.270 align:middle line:84% d is some positive real number such that A here 00:31:14.270 --> 00:31:21.205 align:middle line:84% is contained in the ball of radius d centered at x. 00:31:21.205 --> 00:31:25.976 align:middle line:90% 00:31:25.976 --> 00:31:30.980 align:middle line:84% Now with this-- so there's a few things like this 00:31:30.980 --> 00:31:34.770 align:middle line:90% that we'll see for these things. 00:31:34.770 --> 00:31:39.500 align:middle line:84% The empty set-- if you take x and you take 00:31:39.500 --> 00:31:43.255 align:middle line:84% the subset that has no elements, that's the empty set. 00:31:43.255 --> 00:31:45.830 align:middle line:90% 00:31:45.830 --> 00:31:49.190 align:middle line:84% The empty set is contained in everything. 00:31:49.190 --> 00:31:52.190 align:middle line:84% So actually, the empty set would be bounded. 00:31:52.190 --> 00:31:57.920 align:middle line:84% But it's not a particularly interesting example 00:31:57.920 --> 00:31:58.910 align:middle line:90% of a bounded set. 00:31:58.910 --> 00:32:03.340 align:middle line:84% And it may be that the whole metric space is possible 00:32:03.340 --> 00:32:06.010 align:middle line:84% that the whole metric space is bounded 00:32:06.010 --> 00:32:09.110 align:middle line:90% if it was like an interval. 00:32:09.110 --> 00:32:12.230 align:middle line:84% If it was just an interval where both-- 00:32:12.230 --> 00:32:14.500 align:middle line:84% so suppose you have some interval from A 00:32:14.500 --> 00:32:18.580 align:middle line:90% to B. Then this thing here is-- 00:32:18.580 --> 00:32:22.480 align:middle line:84% if you think about the usual distance on that interval, 00:32:22.480 --> 00:32:25.120 align:middle line:90% then that's a bounded set. 00:32:25.120 --> 00:32:27.010 align:middle line:90% The whole set x is bounded. 00:32:27.010 --> 00:32:32.230 align:middle line:84% So in particular, any subset would also be bounded. 00:32:32.230 --> 00:32:40.300 align:middle line:84% So now let me just prove one theorem, one lemma, that 00:32:40.300 --> 00:32:44.930 align:middle line:84% is another thing that generalizes from R. 00:32:44.930 --> 00:32:48.230 align:middle line:84% And that is that if you take a Cauchy sequence-- 00:32:48.230 --> 00:32:51.730 align:middle line:84% so any Cauchy sequence is bounded. 00:32:51.730 --> 00:32:53.265 align:middle line:90% So let's prove this. 00:32:53.265 --> 00:32:57.770 align:middle line:90% 00:32:57.770 --> 00:33:05.750 align:middle line:84% So we have, again, a metric space. 00:33:05.750 --> 00:33:08.755 align:middle line:90% xn is a Cauchy sequence. 00:33:08.755 --> 00:33:16.130 align:middle line:90% 00:33:16.130 --> 00:33:23.990 align:middle line:84% And then we think about the set xn minus-- 00:33:23.990 --> 00:33:27.870 align:middle line:90% so this is this subset of x. 00:33:27.870 --> 00:33:32.940 align:middle line:84% And I claim that this subset here, this subset here-- 00:33:32.940 --> 00:33:35.965 align:middle line:90% so A here is bounded. 00:33:35.965 --> 00:33:40.760 align:middle line:90% 00:33:40.760 --> 00:33:43.923 align:middle line:90% Now why is that the case? 00:33:43.923 --> 00:33:49.760 align:middle line:90% 00:33:49.760 --> 00:34:04.980 align:middle line:84% So since xn is a sequence, if I choose epsilon with-- 00:34:04.980 --> 00:34:10.770 align:middle line:84% so with epsilon, say, equal to 1, 00:34:10.770 --> 00:34:22.920 align:middle line:84% you have that there exists capital N so that if little n-- 00:34:22.920 --> 00:34:29.520 align:middle line:84% such that if little n and little m 00:34:29.520 --> 00:34:34.050 align:middle line:84% bigger than capital N, then the distance between the two 00:34:34.050 --> 00:34:39.760 align:middle line:90% is smaller than 1. 00:34:39.760 --> 00:34:42.210 align:middle line:84% This is just using that as a Cauchy sequence 00:34:42.210 --> 00:34:48.179 align:middle line:84% and just using one particular epsilon, namely taking epsilon 00:34:48.179 --> 00:34:51.360 align:middle line:90% to be equal to 1. 00:34:51.360 --> 00:34:53.850 align:middle line:90% So now, of course, this means-- 00:34:53.850 --> 00:34:57.720 align:middle line:84% and now you don't need this event for all of this. 00:34:57.720 --> 00:35:00.180 align:middle line:90% You don't need it like this. 00:35:00.180 --> 00:35:06.830 align:middle line:84% So in particular, therefore, as long 00:35:06.830 --> 00:35:11.210 align:middle line:84% as n is bigger than capital N, well, you can use one-- 00:35:11.210 --> 00:35:13.550 align:middle line:90% or you can use m here. 00:35:13.550 --> 00:35:17.130 align:middle line:90% You can use capital N for m. 00:35:17.130 --> 00:35:19.160 align:middle line:84% So then this here-- say, therefore, 00:35:19.160 --> 00:35:26.690 align:middle line:84% for little n bigger than this, we have that this thing here, 00:35:26.690 --> 00:35:29.570 align:middle line:84% the distance to this is smaller than 1. 00:35:29.570 --> 00:35:33.620 align:middle line:90% 00:35:33.620 --> 00:35:41.870 align:middle line:84% And so now you have that if you now 00:35:41.870 --> 00:35:52.380 align:middle line:84% set r equal to 1 plus the maximum of the distance 00:35:52.380 --> 00:36:07.110 align:middle line:84% from xN, x capital N, to xi, where i here runs 00:36:07.110 --> 00:36:11.410 align:middle line:90% from 1 of 2 capital N minus 1. 00:36:11.410 --> 00:36:17.310 align:middle line:84% So there are n minus 1 of these that you take the maximum over. 00:36:17.310 --> 00:36:22.710 align:middle line:84% So if you define little r to be like this, then obviously, 00:36:22.710 --> 00:36:24.700 align:middle line:84% little r is some positive number. 00:36:24.700 --> 00:36:25.830 align:middle line:90% It's finite. 00:36:25.830 --> 00:36:31.620 align:middle line:90% 00:36:31.620 --> 00:36:37.170 align:middle line:84% And now I claim that our set A here is contained in this ball 00:36:37.170 --> 00:36:39.850 align:middle line:90% here around xn. 00:36:39.850 --> 00:36:42.900 align:middle line:90% So I claim this thing here. 00:36:42.900 --> 00:36:57.190 align:middle line:90% 00:36:57.190 --> 00:36:58.070 align:middle line:90% Now let's see. 00:36:58.070 --> 00:37:00.580 align:middle line:90% So why is this the case? 00:37:00.580 --> 00:37:06.050 align:middle line:90% So if x here-- 00:37:06.050 --> 00:37:08.830 align:middle line:90% so we need to prove that-- 00:37:08.830 --> 00:37:24.070 align:middle line:84% we need to show that for all n you have 00:37:24.070 --> 00:37:31.840 align:middle line:84% that the distance here is less than r. 00:37:31.840 --> 00:37:34.870 align:middle line:84% If that's the case, then the whole set 00:37:34.870 --> 00:37:39.310 align:middle line:90% here would lie in this ball. 00:37:39.310 --> 00:37:44.140 align:middle line:90% But you see that if n here-- 00:37:44.140 --> 00:37:50.470 align:middle line:84% well, if n is bigger or equal to capital N, but this-- 00:37:50.470 --> 00:38:01.440 align:middle line:90% 00:38:01.440 --> 00:38:03.250 align:middle line:90% so we need to show this in here. 00:38:03.250 --> 00:38:06.180 align:middle line:84% If little n is bigger than capital N, 00:38:06.180 --> 00:38:11.730 align:middle line:84% then we have that the distance from this little n to this one 00:38:11.730 --> 00:38:13.960 align:middle line:90% here by the tail-- 00:38:13.960 --> 00:38:16.320 align:middle line:90% the tail was less than 1. 00:38:16.320 --> 00:38:21.510 align:middle line:84% But this thing here is actually why-- this is actually-- 00:38:21.510 --> 00:38:25.930 align:middle line:84% this is equal to r because it was 1 plus this maximum. 00:38:25.930 --> 00:38:29.170 align:middle line:84% And then this is maximum over nonnegative numbers. 00:38:29.170 --> 00:38:36.720 align:middle line:84% So you see that for the whole tail, we have this property. 00:38:36.720 --> 00:38:43.150 align:middle line:84% But for all the other ones, for n less than 00:38:43.150 --> 00:38:50.440 align:middle line:84% or equal less than n minus 1, these are the remaining ones. 00:38:50.440 --> 00:39:00.530 align:middle line:84% There, the distance from xn to x capital N, where it's-- 00:39:00.530 --> 00:39:02.810 align:middle line:90% this is one of these guys. 00:39:02.810 --> 00:39:04.870 align:middle line:90% So it's clearly this one here. 00:39:04.870 --> 00:39:09.800 align:middle line:84% It's clearly less than capital R. 00:39:09.800 --> 00:39:11.800 align:middle line:84% And it's really strictly less than 00:39:11.800 --> 00:39:13.460 align:middle line:90% because there's this one also. 00:39:13.460 --> 00:39:14.650 align:middle line:90% Yeah? 00:39:14.650 --> 00:39:17.330 align:middle line:84% AUDIENCE: [INAUDIBLE] does it has to be 1 plus the max, 00:39:17.330 --> 00:39:20.650 align:middle line:84% or could it be the max of 1 or all of those elements 00:39:20.650 --> 00:39:22.810 align:middle line:90% of the sequence? 00:39:22.810 --> 00:39:26.410 align:middle line:84% PROFESSOR: I wanted this one here to be strict. 00:39:26.410 --> 00:39:28.040 align:middle line:90% It's just a silly thing. 00:39:28.040 --> 00:39:30.670 align:middle line:84% But that's why I took it like that. 00:39:30.670 --> 00:39:32.450 align:middle line:84% Because otherwise this would be strict. 00:39:32.450 --> 00:39:38.190 align:middle line:84% But it's just a trivial thing, and it doesn't matter very much. 00:39:38.190 --> 00:39:40.190 align:middle line:84% At this stage, it makes no difference, actually. 00:39:40.190 --> 00:39:43.120 align:middle line:90% But it was just to use this. 00:39:43.120 --> 00:39:45.020 align:middle line:84% We'll talk a little bit more about the balls 00:39:45.020 --> 00:39:50.420 align:middle line:84% because there is another way you could define the ball. 00:39:50.420 --> 00:39:52.480 align:middle line:84% Maybe I'll say a few words about that now. 00:39:52.480 --> 00:39:55.500 align:middle line:90% 00:39:55.500 --> 00:40:00.680 align:middle line:84% So what this proves, again, is that in any metric space, 00:40:00.680 --> 00:40:05.810 align:middle line:84% a Cauchy sequence is automatically bounded. 00:40:05.810 --> 00:40:07.640 align:middle line:84% And so remember, in any metric space, 00:40:07.640 --> 00:40:10.080 align:middle line:84% a convergent sequence is a Cauchy sequence. 00:40:10.080 --> 00:40:13.610 align:middle line:84% So a convergent sequence is also bounded. 00:40:13.610 --> 00:40:18.060 align:middle line:84% And now there's this question, which is a natural question, 00:40:18.060 --> 00:40:22.550 align:middle line:84% and this is connected to the definition of ball. 00:40:22.550 --> 00:40:26.030 align:middle line:84% So the way we define-- and it's actually worthwhile 00:40:26.030 --> 00:40:31.550 align:middle line:84% emphasizing because it's useful in the next thing 00:40:31.550 --> 00:40:34.280 align:middle line:90% that we want to get to. 00:40:34.280 --> 00:40:39.860 align:middle line:84% And so when I define a ball, I assumed-- 00:40:39.860 --> 00:40:45.360 align:middle line:84% so the definition of the ball-- just write it up again here. 00:40:45.360 --> 00:40:48.780 align:middle line:84% So the definition of a ball in a metric space-- 00:40:48.780 --> 00:40:56.550 align:middle line:84% these are the y in this set so that the distance to the center 00:40:56.550 --> 00:40:59.670 align:middle line:90% is strictly less than r. 00:40:59.670 --> 00:41:05.070 align:middle line:84% You could, of course, have defined it differently. 00:41:05.070 --> 00:41:08.550 align:middle line:84% You could define it that this is strictly 00:41:08.550 --> 00:41:11.740 align:middle line:84% less or is there strictly less than or equal to-- 00:41:11.740 --> 00:41:15.540 align:middle line:84% I want to use strictly less here. 00:41:15.540 --> 00:41:19.300 align:middle line:84% But sometimes, if I want to be very clear, 00:41:19.300 --> 00:41:21.970 align:middle line:84% I will talk about open and closed ball. 00:41:21.970 --> 00:41:24.010 align:middle line:84% And this is what we're getting to next. 00:41:24.010 --> 00:41:26.470 align:middle line:84% And so when I'm talking about open ball-- 00:41:26.470 --> 00:41:29.760 align:middle line:84% but I'll try not to use that term here. 00:41:29.760 --> 00:41:33.060 align:middle line:90% But open would always mean this. 00:41:33.060 --> 00:41:37.570 align:middle line:84% Closed ball would mean that it's less than or equal to. 00:41:37.570 --> 00:41:39.210 align:middle line:90% Sorry, yeah? 00:41:39.210 --> 00:41:43.370 align:middle line:84% AUDIENCE: Would you talk to me about the neighborhood example? 00:41:43.370 --> 00:41:46.940 align:middle line:84% Is that reverse-- is x in that case a subset? 00:41:46.940 --> 00:41:48.860 align:middle line:84% PROFESSOR: Yeah, that's right, exactly. 00:41:48.860 --> 00:41:49.710 align:middle line:90% That's right. 00:41:49.710 --> 00:41:52.640 align:middle line:90% 00:41:52.640 --> 00:41:57.110 align:middle line:90% And I didn't actually-- 00:41:57.110 --> 00:41:59.400 align:middle line:84% I already posted the lecture notes. 00:41:59.400 --> 00:42:01.760 align:middle line:84% And so in the lecture notes, I didn't actually 00:42:01.760 --> 00:42:03.360 align:middle line:84% talk about tubular neighborhoods. 00:42:03.360 --> 00:42:09.210 align:middle line:84% But it maybe is worth thinking about. 00:42:09.210 --> 00:42:13.160 align:middle line:90% So you could look at it as-- 00:42:13.160 --> 00:42:15.205 align:middle line:84% again, you have a metric space here. 00:42:15.205 --> 00:42:18.440 align:middle line:90% 00:42:18.440 --> 00:42:21.095 align:middle line:90% And you could have a subset-- 00:42:21.095 --> 00:42:25.850 align:middle line:84% A has a subset, any subset, of the metric space. 00:42:25.850 --> 00:42:29.490 align:middle line:84% And you could define the tubular neighborhood of a subset. 00:42:29.490 --> 00:42:32.460 align:middle line:84% And so you could do this similarly. 00:42:32.460 --> 00:42:37.340 align:middle line:84% So this would be the y in x so that the distance 00:42:37.340 --> 00:42:49.980 align:middle line:84% from y to x here is strictly less than r 00:42:49.980 --> 00:42:59.830 align:middle line:90% for some x in a [INAUDIBLE]. 00:42:59.830 --> 00:43:02.800 align:middle line:90% So this is how-- 00:43:02.800 --> 00:43:07.020 align:middle line:84% so again, if you think about our tree, you take some-- 00:43:07.020 --> 00:43:08.940 align:middle line:84% I mean, maybe we shouldn't talk about-- 00:43:08.940 --> 00:43:13.440 align:middle line:84% maybe we should draw it more crazy, like some curve, maybe. 00:43:13.440 --> 00:43:18.040 align:middle line:84% And if you're looking at the set that have distance at most-- 00:43:18.040 --> 00:43:20.958 align:middle line:84% this is some sort of tube, like this. 00:43:20.958 --> 00:43:24.660 align:middle line:90% 00:43:24.660 --> 00:43:27.220 align:middle line:84% This here would be the [INAUDIBLE]. 00:43:27.220 --> 00:43:30.720 align:middle line:84% And then here, at the end, it would look like a cap, 00:43:30.720 --> 00:43:33.150 align:middle line:90% like this. 00:43:33.150 --> 00:43:35.430 align:middle line:84% This here would be like a tubular neighborhood 00:43:35.430 --> 00:43:36.520 align:middle line:90% of this curve. 00:43:36.520 --> 00:43:39.280 align:middle line:90% 00:43:39.280 --> 00:43:44.950 align:middle line:84% And so, as you're saying that in the case 00:43:44.950 --> 00:43:49.730 align:middle line:84% where A here is just a point, then those two coincide. 00:43:49.730 --> 00:43:53.060 align:middle line:90% 00:43:53.060 --> 00:43:57.370 align:middle line:84% So now one of the key things when 00:43:57.370 --> 00:44:00.760 align:middle line:84% you're thinking about convergence in R, 00:44:00.760 --> 00:44:04.810 align:middle line:84% remember that this stuff, when we use this time and time again, 00:44:04.810 --> 00:44:06.220 align:middle line:90% that it was often-- 00:44:06.220 --> 00:44:09.940 align:middle line:84% we saw this for series, that it was often 00:44:09.940 --> 00:44:15.200 align:middle line:84% difficult to determine the exact limit of a series. 00:44:15.200 --> 00:44:19.550 align:middle line:84% But the question was always, does it converge? 00:44:19.550 --> 00:44:26.410 align:middle line:84% And so if it converged, if it converged, then often, I 00:44:26.410 --> 00:44:29.320 align:middle line:84% mean, also, if you would have using this in some 00:44:29.320 --> 00:44:34.210 align:middle line:84% of application, it's often in all kind of fields 00:44:34.210 --> 00:44:37.890 align:middle line:84% that it comes up that things are written in terms of a series. 00:44:37.890 --> 00:44:41.850 align:middle line:84% And it's not clear what it's converting to. 00:44:41.850 --> 00:44:43.920 align:middle line:84% But you want to know whether it's converting. 00:44:43.920 --> 00:44:45.990 align:middle line:84% And then you also want to-- so you 00:44:45.990 --> 00:44:50.460 align:middle line:84% want to estimate what is the limit within a certain error 00:44:50.460 --> 00:44:51.070 align:middle line:90% term. 00:44:51.070 --> 00:44:53.640 align:middle line:84% And you can use some of these comparison tests 00:44:53.640 --> 00:44:59.700 align:middle line:84% to determine-- to estimate how close you 00:44:59.700 --> 00:45:01.510 align:middle line:90% are to the actual limit. 00:45:01.510 --> 00:45:07.260 align:middle line:84% But in any case, the thing is it's very useful 00:45:07.260 --> 00:45:11.820 align:middle line:84% to have a criteria that is easy to work with 00:45:11.820 --> 00:45:16.170 align:middle line:84% and that say that it's converging without having 00:45:16.170 --> 00:45:18.100 align:middle line:90% to be able to find the limits. 00:45:18.100 --> 00:45:21.610 align:middle line:84% And like the Cauchy sequence, this was like a notion of that. 00:45:21.610 --> 00:45:30.450 align:middle line:84% And so again, the key on R, the key for using that 00:45:30.450 --> 00:45:34.510 align:middle line:84% was the Bolzano-Weierstrass theorem. 00:45:34.510 --> 00:45:39.190 align:middle line:84% So this was the one that was the key for using to get 00:45:39.190 --> 00:45:41.690 align:middle line:90% the quasi convergence theorem. 00:45:41.690 --> 00:45:43.620 align:middle line:84% So the Bolzano-Weierstrass theorem-- 00:45:43.620 --> 00:45:52.660 align:middle line:90% 00:45:52.660 --> 00:45:58.570 align:middle line:84% so remember that this is for any [INAUDIBLE] 00:45:58.570 --> 00:46:11.240 align:middle line:84% that any bounded sequence has a convergent subsequence. 00:46:11.240 --> 00:46:22.390 align:middle line:90% 00:46:22.390 --> 00:46:25.270 align:middle line:84% That was Bolzano-Weierstrass theorem. 00:46:25.270 --> 00:46:29.440 align:middle line:84% In a general metric space, it makes sense 00:46:29.440 --> 00:46:35.510 align:middle line:84% to talk about a bounded sequence. 00:46:35.510 --> 00:46:39.020 align:middle line:84% But it's not always the case that a bounded sequence 00:46:39.020 --> 00:46:40.830 align:middle line:90% has a convergent subsequence. 00:46:40.830 --> 00:46:42.570 align:middle line:90% So it does not hold. 00:46:42.570 --> 00:46:44.070 align:middle line:90% The Bolzano does not hold. 00:46:44.070 --> 00:46:51.920 align:middle line:84% So warning, Bolzano-Weierstrass does not 00:46:51.920 --> 00:46:59.200 align:middle line:90% hold in a general metric space. 00:46:59.200 --> 00:47:08.600 align:middle line:90% 00:47:08.600 --> 00:47:15.050 align:middle line:84% And in fact, of course, the example that we looked at, 00:47:15.050 --> 00:47:18.000 align:middle line:84% where the example was x here, like this, 00:47:18.000 --> 00:47:20.010 align:middle line:84% and when we're looking at that sequence, 00:47:20.010 --> 00:47:22.820 align:middle line:90% this is a bounded sequence. 00:47:22.820 --> 00:47:27.150 align:middle line:84% But it's bounded, but it doesn't have a convergent subsequence. 00:47:27.150 --> 00:47:28.980 align:middle line:90% It's actually a Cauchy sequence. 00:47:28.980 --> 00:47:30.110 align:middle line:90% But it doesn't [INAUDIBLE]. 00:47:30.110 --> 00:47:34.360 align:middle line:84% So it doesn't hold in a general metric space, 00:47:34.360 --> 00:47:51.960 align:middle line:84% but it does hold if the metric space is compact. 00:47:51.960 --> 00:47:54.850 align:middle line:90% 00:47:54.850 --> 00:47:57.320 align:middle line:84% And so that's what I want to talk about next. 00:47:57.320 --> 00:48:01.480 align:middle line:84% But in order to talk about what it means for metric space 00:48:01.480 --> 00:48:09.610 align:middle line:84% to be compact, then I need to talk about-- 00:48:09.610 --> 00:48:15.700 align:middle line:84% I need to define what it means for a set to be open. 00:48:15.700 --> 00:48:36.500 align:middle line:84% So we need the notion of an open set, an open subset 00:48:36.500 --> 00:48:42.340 align:middle line:90% of a metric space. 00:48:42.340 --> 00:48:49.520 align:middle line:90% 00:48:49.520 --> 00:48:54.530 align:middle line:90% So we have a metric space again. 00:48:54.530 --> 00:49:00.050 align:middle line:90% And so A here is said to be-- 00:49:00.050 --> 00:49:02.610 align:middle line:90% let's call it O for open. 00:49:02.610 --> 00:49:21.470 align:middle line:84% O is said to be an open subset if for all x in O-- 00:49:21.470 --> 00:49:28.440 align:middle line:84% so the idea is that you have here, this here is O. 00:49:28.440 --> 00:49:34.410 align:middle line:90% So this is maybe O. 00:49:34.410 --> 00:49:37.770 align:middle line:84% And so the idea is that for each element, 00:49:37.770 --> 00:49:41.920 align:middle line:84% there exists some ball centered at that point. 00:49:41.920 --> 00:49:47.090 align:middle line:84% So for each x for all x in O, there exists. 00:49:47.090 --> 00:49:52.080 align:middle line:90% 00:49:52.080 --> 00:49:55.410 align:middle line:84% r greater than 0-- it may be incredibly small-- 00:49:55.410 --> 00:50:02.310 align:middle line:84% so that not just x but the whole ball around x 00:50:02.310 --> 00:50:04.180 align:middle line:90% is contained in the open set. 00:50:04.180 --> 00:50:07.710 align:middle line:84% So you see that not just this point here, 00:50:07.710 --> 00:50:12.510 align:middle line:84% but the whole ball here would have to be contained in O. 00:50:12.510 --> 00:50:18.040 align:middle line:84% So in this picture here, when I'm illustrating this, 00:50:18.040 --> 00:50:22.330 align:middle line:84% I'm thinking about x as being R2. 00:50:22.330 --> 00:50:24.210 align:middle line:84% And I'm thinking about the distance 00:50:24.210 --> 00:50:25.870 align:middle line:84% being the usual Euclidean distance. 00:50:25.870 --> 00:50:28.640 align:middle line:90% 00:50:28.640 --> 00:50:31.080 align:middle line:84% But then, actually, if this is open, 00:50:31.080 --> 00:50:35.660 align:middle line:84% it means that you cannot include these points here on this curve. 00:50:35.660 --> 00:50:38.220 align:middle line:84% They would not be included in the set 00:50:38.220 --> 00:50:42.960 align:middle line:84% because if you take any point on this boundary here, so to speak, 00:50:42.960 --> 00:50:48.690 align:middle line:84% then a ball part of it would be outside the set. 00:50:48.690 --> 00:50:51.860 align:middle line:84% So if this really was open, then this curve 00:50:51.860 --> 00:50:53.550 align:middle line:90% here was not included. 00:50:53.550 --> 00:50:56.090 align:middle line:84% And you see that as you're getting closer and closer 00:50:56.090 --> 00:51:00.800 align:middle line:84% to that curve, you can still have a little ball that 00:51:00.800 --> 00:51:02.550 align:middle line:90% is inside the set. 00:51:02.550 --> 00:51:05.690 align:middle line:84% But as you're getting closer, the radius of this little ball 00:51:05.690 --> 00:51:07.610 align:middle line:90% have to be smaller. 00:51:07.610 --> 00:51:10.160 align:middle line:90% But that's fine. 00:51:10.160 --> 00:51:15.860 align:middle line:84% So this is what it means for a subset to be open. 00:51:15.860 --> 00:51:21.380 align:middle line:84% And the first thing that you observe 00:51:21.380 --> 00:51:23.790 align:middle line:90% that with this definition-- 00:51:23.790 --> 00:51:26.680 align:middle line:84% there are always two open subsets. 00:51:26.680 --> 00:51:31.410 align:middle line:90% So again, this thing here-- 00:51:31.410 --> 00:51:38.620 align:middle line:90% so note that the empty set-- 00:51:38.620 --> 00:51:39.970 align:middle line:90% but there's no condition. 00:51:39.970 --> 00:51:43.810 align:middle line:84% If there was an empty set, say for all elements in the set, 00:51:43.810 --> 00:51:47.740 align:middle line:84% But there is no element, so there's nothing to be satisfied. 00:51:47.740 --> 00:51:53.045 align:middle line:84% So the empty set and all of x are open subsets. 00:51:53.045 --> 00:51:59.940 align:middle line:90% 00:51:59.940 --> 00:52:03.160 align:middle line:90% So do note that. 00:52:03.160 --> 00:52:15.750 align:middle line:84% So in particular, if so if x here was not the whole real line 00:52:15.750 --> 00:52:19.410 align:middle line:84% but it was just the interval from 0 to 1, 00:52:19.410 --> 00:52:23.170 align:middle line:84% and then the distance between elements 00:52:23.170 --> 00:52:26.800 align:middle line:84% was the usual distance, then, actually, 00:52:26.800 --> 00:52:32.080 align:middle line:84% even the closed interval, which is the whole set, would be open. 00:52:32.080 --> 00:52:37.960 align:middle line:84% So you just should be careful with what the set is-- 00:52:37.960 --> 00:52:41.740 align:middle line:90% 00:52:41.740 --> 00:52:43.310 align:middle line:90% what set [INAUDIBLE]. 00:52:43.310 --> 00:52:50.140 align:middle line:90% 00:52:50.140 --> 00:52:53.090 align:middle line:90% So now the key here is that-- 00:52:53.090 --> 00:53:01.720 align:middle line:84% so if you have an open set, how do you construct a-- 00:53:01.720 --> 00:53:04.060 align:middle line:84% well, maybe the first thing I should do 00:53:04.060 --> 00:53:11.800 align:middle line:84% is I should prove a lemma that is the following. 00:53:11.800 --> 00:53:15.610 align:middle line:84% And then, after that, we will talk about how 00:53:15.610 --> 00:53:21.020 align:middle line:90% to get lots of open sets. 00:53:21.020 --> 00:53:23.650 align:middle line:90% 00:53:23.650 --> 00:53:25.090 align:middle line:90% So we have two examples. 00:53:25.090 --> 00:53:29.370 align:middle line:84% At the moment, we have two examples of-- 00:53:29.370 --> 00:53:32.010 align:middle line:84% so if you take a metric space, at the moment, 00:53:32.010 --> 00:53:35.250 align:middle line:84% we know that there's always two subsets that 00:53:35.250 --> 00:53:38.160 align:middle line:90% are open, the empty set and x. 00:53:38.160 --> 00:53:40.140 align:middle line:84% But they're not very interesting. 00:53:40.140 --> 00:53:45.360 align:middle line:84% So a more interesting example is that if you take a metric space 00:53:45.360 --> 00:53:47.050 align:middle line:90% and you take a ball-- 00:53:47.050 --> 00:53:50.410 align:middle line:84% and now it's crucial how we define the ball. 00:53:50.410 --> 00:53:56.010 align:middle line:84% And so it's crucial that this inequality is strict. 00:53:56.010 --> 00:54:02.520 align:middle line:84% So for a metric space x here is in x. 00:54:02.520 --> 00:54:06.730 align:middle line:90% It could be any element in x. 00:54:06.730 --> 00:54:12.250 align:middle line:84% This thing here, r here, is strictly positive. 00:54:12.250 --> 00:54:15.310 align:middle line:84% It also has to be strictly positive. 00:54:15.310 --> 00:54:18.430 align:middle line:90% Then this thing here is open. 00:54:18.430 --> 00:54:25.030 align:middle line:84% Any metric ball like this is an open subset. 00:54:25.030 --> 00:54:26.480 align:middle line:90% Let's try to prove that. 00:54:26.480 --> 00:54:28.300 align:middle line:84% And again, whether you're proving things 00:54:28.300 --> 00:54:32.410 align:middle line:84% like this for a metric space, the only nontrivial thing 00:54:32.410 --> 00:54:34.250 align:middle line:90% is the triangle inequality. 00:54:34.250 --> 00:54:37.870 align:middle line:84% So this is relying on the triangle inequality. 00:54:37.870 --> 00:54:41.890 align:middle line:84% And so for a lot of things, when you're 00:54:41.890 --> 00:54:44.540 align:middle line:84% proving things about a metric space, 00:54:44.540 --> 00:54:49.840 align:middle line:84% it's often quite helpful to draw a little picture. 00:54:49.840 --> 00:54:54.370 align:middle line:84% But of course, you have to be careful with, again, 00:54:54.370 --> 00:54:59.170 align:middle line:84% then the intuition comes from Rn. 00:54:59.170 --> 00:55:03.940 align:middle line:84% And sometimes, the metric is a little bit more complicated 00:55:03.940 --> 00:55:04.840 align:middle line:90% than that. 00:55:04.840 --> 00:55:10.730 align:middle line:84% So this here, imagine that this is the ball of radius r. 00:55:10.730 --> 00:55:13.480 align:middle line:90% Here you have some x. 00:55:13.480 --> 00:55:17.000 align:middle line:90% Now take an element y. 00:55:17.000 --> 00:55:22.700 align:middle line:84% So suppose that y here is now in this ball here. 00:55:22.700 --> 00:55:33.860 align:middle line:84% Then we want to show that there exists 00:55:33.860 --> 00:55:42.620 align:middle line:84% that for some s bigger than 0, the ball here of radius 00:55:42.620 --> 00:55:48.530 align:middle line:84% s around y is now contained in this ball here. 00:55:48.530 --> 00:55:50.940 align:middle line:90% So here, you have y. 00:55:50.940 --> 00:55:52.760 align:middle line:84% And we want to prove that there is 00:55:52.760 --> 00:55:59.570 align:middle line:84% some little ball around y that is 00:55:59.570 --> 00:56:02.270 align:middle line:90% contained in the original ball. 00:56:02.270 --> 00:56:08.690 align:middle line:84% And so, again, this is going to rely on the triangle inequality. 00:56:08.690 --> 00:56:11.760 align:middle line:90% So you have y here. 00:56:11.760 --> 00:56:14.620 align:middle line:84% y was supposed to be in this ball here. 00:56:14.620 --> 00:56:19.780 align:middle line:84% This means that the distance from x to y 00:56:19.780 --> 00:56:23.620 align:middle line:90% is strictly the s in r. 00:56:23.620 --> 00:56:31.600 align:middle line:84% You take now s here to be r minus this distance here. 00:56:31.600 --> 00:56:34.000 align:middle line:90% Why do I do this? 00:56:34.000 --> 00:56:36.910 align:middle line:84% In my picture here, I have here x. 00:56:36.910 --> 00:56:39.250 align:middle line:90% I have here y. 00:56:39.250 --> 00:56:46.840 align:middle line:84% And our intuition is that if we go from x and we go through y 00:56:46.840 --> 00:56:49.300 align:middle line:84% and then we go straight to the boundary, 00:56:49.300 --> 00:56:55.160 align:middle line:84% then this thing here is dx comma y. 00:56:55.160 --> 00:57:01.700 align:middle line:90% And this one here is r minus d. 00:57:01.700 --> 00:57:04.870 align:middle line:84% This is the remaining-- this whole here is r. 00:57:04.870 --> 00:57:07.100 align:middle line:90% This is dx comma y. 00:57:07.100 --> 00:57:10.750 align:middle line:90% So this here must be this here. 00:57:10.750 --> 00:57:14.330 align:middle line:84% And so if you're setting this equal to s, 00:57:14.330 --> 00:57:18.260 align:middle line:84% and now you imagine taking a ball here, well, then 00:57:18.260 --> 00:57:20.960 align:middle line:84% at least on the picture, this whole ball 00:57:20.960 --> 00:57:23.240 align:middle line:84% is contained in the original ball. 00:57:23.240 --> 00:57:27.390 align:middle line:84% So let's try to see if that works. 00:57:27.390 --> 00:57:29.900 align:middle line:84% If that works, then it really only used 00:57:29.900 --> 00:57:32.180 align:middle line:90% the triangle inequality. 00:57:32.180 --> 00:57:34.770 align:middle line:84% So we're setting s equal to this. 00:57:34.770 --> 00:57:39.900 align:middle line:84% And then we're looking at this ball here around S. 00:57:39.900 --> 00:57:44.130 align:middle line:84% And we take an element here in this ball. 00:57:44.130 --> 00:57:46.260 align:middle line:90% And we now need to prove-- 00:57:46.260 --> 00:57:47.650 align:middle line:90% so we need to show. 00:57:47.650 --> 00:57:53.210 align:middle line:90% 00:57:53.210 --> 00:57:55.640 align:middle line:84% We want to prove that this whole ball is 00:57:55.640 --> 00:57:57.150 align:middle line:90% contained in the original ball. 00:57:57.150 --> 00:58:01.160 align:middle line:84% So this means that we need to prove that for any such element 00:58:01.160 --> 00:58:06.170 align:middle line:90% set here is in this thing here. 00:58:06.170 --> 00:58:07.600 align:middle line:90% That's what we need to prove. 00:58:07.600 --> 00:58:14.220 align:middle line:90% 00:58:14.220 --> 00:58:16.025 align:middle line:90% And so now you see-- 00:58:16.025 --> 00:58:23.590 align:middle line:90% 00:58:23.590 --> 00:58:26.650 align:middle line:90% so we take z. 00:58:26.650 --> 00:58:29.020 align:middle line:84% So we're going to look at this distance. 00:58:29.020 --> 00:58:31.530 align:middle line:84% In order to prove this, we need to prove 00:58:31.530 --> 00:58:35.920 align:middle line:84% that the distance from z to x is strictly less than r. 00:58:35.920 --> 00:58:38.520 align:middle line:90% So let's examine that distance. 00:58:38.520 --> 00:58:43.110 align:middle line:84% Triangle inequality, say, that this is bounded by the distance 00:58:43.110 --> 00:58:49.890 align:middle line:84% from x to y plus the distance from y to z. 00:58:49.890 --> 00:58:58.930 align:middle line:84% This one here from y to z lie in the ball of radius s around y. 00:58:58.930 --> 00:59:01.480 align:middle line:84% So this means that this thing here, 00:59:01.480 --> 00:59:06.180 align:middle line:84% this thing here is strictly less than s. 00:59:06.180 --> 00:59:13.760 align:middle line:84% And so you see that this thing here is now less than d x, 00:59:13.760 --> 00:59:19.280 align:middle line:90% y plus s. 00:59:19.280 --> 00:59:23.895 align:middle line:84% Now you fill in what s is, this here. 00:59:23.895 --> 00:59:30.060 align:middle line:90% s was equal to r minus d x, y. 00:59:30.060 --> 00:59:31.680 align:middle line:90% That was the definition of s. 00:59:31.680 --> 00:59:34.100 align:middle line:84% And so now you see that this thing here 00:59:34.100 --> 00:59:37.830 align:middle line:84% is r because those two things cancel out. 00:59:37.830 --> 00:59:40.280 align:middle line:84% And you see that this here guarantee you 00:59:40.280 --> 00:59:42.650 align:middle line:90% that the inequality was strict. 00:59:42.650 --> 00:59:47.150 align:middle line:84% And so it really is that whole ball 00:59:47.150 --> 00:59:52.850 align:middle line:84% was-- so we Conclude that this whole ball here was contained 00:59:52.850 --> 00:59:55.080 align:middle line:90% in the original form. 00:59:55.080 --> 01:00:01.130 align:middle line:84% And again, the proof of it was just from this picture 01:00:01.130 --> 01:00:04.730 align:middle line:90% that you make kind of rigorous. 01:00:04.730 --> 01:00:08.190 align:middle line:84% And the only nontrivial things at the moment we really 01:00:08.190 --> 01:00:11.563 align:middle line:84% have in a metric space is a triangle in concern 01:00:11.563 --> 01:00:13.355 align:middle line:84% because we haven't built up a lot of stuff. 01:00:13.355 --> 01:00:17.460 align:middle line:90% 01:00:17.460 --> 01:00:20.580 align:middle line:84% So you see, this immediately gives you 01:00:20.580 --> 01:00:29.470 align:middle line:84% a lot of example of open subsets of a general metric space. 01:00:29.470 --> 01:00:33.390 align:middle line:84% And if we had looked at exactly the same stuff-- 01:00:33.390 --> 01:00:36.930 align:middle line:84% tubular neighborhood-- we would find that those are also open 01:00:36.930 --> 01:00:37.620 align:middle line:90% subsets. 01:00:37.620 --> 01:00:38.670 align:middle line:90% Yeah? 01:00:38.670 --> 01:00:40.570 align:middle line:84% AUDIENCE: Also we look at subsets, 01:00:40.570 --> 01:00:42.172 align:middle line:90% or do those have any value? 01:00:42.172 --> 01:00:42.880 align:middle line:90% PROFESSOR: Close? 01:00:42.880 --> 01:00:45.000 align:middle line:90% Yeah, very much so. 01:00:45.000 --> 01:00:49.140 align:middle line:84% But we haven't defined what it means to be close. 01:00:49.140 --> 01:00:54.870 align:middle line:84% But you're right, and we will do this. 01:00:54.870 --> 01:00:59.400 align:middle line:84% So at the moment, we don't know what a close thing is. 01:00:59.400 --> 01:01:01.810 align:middle line:90% And maybe I will just define it. 01:01:01.810 --> 01:01:04.510 align:middle line:90% And then we can-- 01:01:04.510 --> 01:01:08.590 align:middle line:90% 01:01:08.590 --> 01:01:12.010 align:middle line:90% so let's define. 01:01:12.010 --> 01:01:16.030 align:middle line:84% But I'll talk more about this next time-- 01:01:16.030 --> 01:01:17.590 align:middle line:90% but now when you're asking. 01:01:17.590 --> 01:01:20.480 align:middle line:90% 01:01:20.480 --> 01:01:25.630 align:middle line:84% So we have x here is a metric space again. 01:01:25.630 --> 01:01:32.483 align:middle line:84% And then see here, it's a subset of x is said to be closed. 01:01:32.483 --> 01:01:41.350 align:middle line:90% 01:01:41.350 --> 01:01:44.500 align:middle line:90% And then now it may-- 01:01:44.500 --> 01:01:45.980 align:middle line:90% seems almost cheating. 01:01:45.980 --> 01:01:49.246 align:middle line:84% It's said to be closed if the complement-- 01:01:49.246 --> 01:01:54.430 align:middle line:90% 01:01:54.430 --> 01:01:58.375 align:middle line:84% so this means that the complement of C-- 01:01:58.375 --> 01:02:01.480 align:middle line:90% 01:02:01.480 --> 01:02:17.800 align:middle line:84% so the complement of C-- these are all the x in x 01:02:17.800 --> 01:02:24.930 align:middle line:90% so that x is not in C. 01:02:24.930 --> 01:02:28.290 align:middle line:84% So the complement a set is everything 01:02:28.290 --> 01:02:31.360 align:middle line:90% in x that is not in that set. 01:02:31.360 --> 01:02:39.270 align:middle line:84% And a set is said to be closed if the complement is open. 01:02:39.270 --> 01:02:43.320 align:middle line:84% And so now, once you have this definition 01:02:43.320 --> 01:02:47.340 align:middle line:84% and now prove this next time, that if you're 01:02:47.340 --> 01:02:52.110 align:middle line:84% looking at what I denote like this, 01:02:52.110 --> 01:03:01.120 align:middle line:84% so this is the elements where the distance to x 01:03:01.120 --> 01:03:05.540 align:middle line:84% is less than or equal to-- so this is not any more strict. 01:03:05.540 --> 01:03:09.543 align:middle line:84% This thing here is then a closed set. 01:03:09.543 --> 01:03:10.960 align:middle line:84% AUDIENCE: Could you also define it 01:03:10.960 --> 01:03:16.360 align:middle line:84% by the way we define an open subset but saying that R instead 01:03:16.360 --> 01:03:17.448 align:middle line:90% of being strictly? 01:03:17.448 --> 01:03:19.365 align:middle line:84% PROFESSOR: No, I couldn't define it like that. 01:03:19.365 --> 01:03:23.110 align:middle line:90% 01:03:23.110 --> 01:03:25.010 align:middle line:84% So I mean, you could define it other ways. 01:03:25.010 --> 01:03:27.218 align:middle line:84% There's another standard where you would talk about-- 01:03:27.218 --> 01:03:30.400 align:middle line:84% there's another equivalent to say that if you take-- 01:03:30.400 --> 01:03:32.710 align:middle line:84% but I don't want to get into that because-- 01:03:32.710 --> 01:03:37.690 align:middle line:84% but there's another way we get to this. 01:03:37.690 --> 01:03:41.270 align:middle line:84% And closed sets, we really mostly talk about next time. 01:03:41.270 --> 01:03:44.230 align:middle line:84% It was just because you were asking. 01:03:44.230 --> 01:03:46.550 align:middle line:90% But let me just talk about-- 01:03:46.550 --> 01:03:50.800 align:middle line:84% so this is one, but I'll just repeat this in just a second. 01:03:50.800 --> 01:03:56.155 align:middle line:84% So in order to produce more sets, so we start with-- 01:03:56.155 --> 01:03:59.310 align:middle line:90% 01:03:59.310 --> 01:04:02.180 align:middle line:84% actually, I should just note, although just now when 01:04:02.180 --> 01:04:04.740 align:middle line:84% I talk about, just the definition, 01:04:04.740 --> 01:04:09.560 align:middle line:84% I should just immediately note that there are always two closed 01:04:09.560 --> 01:04:14.780 align:middle line:84% sets because you see, the empty set was always open, 01:04:14.780 --> 01:04:17.990 align:middle line:84% and the full set was always open. 01:04:17.990 --> 01:04:22.320 align:middle line:84% And so the complement of the empty set, that's everything. 01:04:22.320 --> 01:04:24.630 align:middle line:90% So that's closed. 01:04:24.630 --> 01:04:27.750 align:middle line:84% And the complement of everything is nothing. 01:04:27.750 --> 01:04:28.650 align:middle line:90% It's empty set. 01:04:28.650 --> 01:04:29.910 align:middle line:90% So that's also closed. 01:04:29.910 --> 01:04:32.390 align:middle line:84% So those two sets are always closed. 01:04:32.390 --> 01:04:34.050 align:middle line:90% But let me just talk about-- 01:04:34.050 --> 01:04:40.790 align:middle line:84% so I want to think about how do you produce open sets? 01:04:40.790 --> 01:04:46.580 align:middle line:84% So how do you know that there are a lot of open sets? 01:04:46.580 --> 01:04:50.460 align:middle line:84% And so if you take sets, if you take-- 01:04:50.460 --> 01:05:03.735 align:middle line:84% so this is like operations on sets on subsets of a given set. 01:05:03.735 --> 01:05:07.570 align:middle line:90% 01:05:07.570 --> 01:05:11.270 align:middle line:84% So here, I'm thinking about just x as being a set. 01:05:11.270 --> 01:05:14.530 align:middle line:84% It does not need to be a metric space. 01:05:14.530 --> 01:05:21.030 align:middle line:84% So if I take a set then and I take two elements, two subsets. 01:05:21.030 --> 01:05:23.680 align:middle line:90% 01:05:23.680 --> 01:05:25.820 align:middle line:90% So these are two subsets. 01:05:25.820 --> 01:05:30.220 align:middle line:84% Then I can form the union of these two. 01:05:30.220 --> 01:05:31.545 align:middle line:90% This is the union. 01:05:31.545 --> 01:05:36.580 align:middle line:90% 01:05:36.580 --> 01:05:39.380 align:middle line:90% And the union here of two sets-- 01:05:39.380 --> 01:05:41.770 align:middle line:84% so the union-- and again, I denote it 01:05:41.770 --> 01:05:45.850 align:middle line:84% like this, [INAUDIBLE], like this. 01:05:45.850 --> 01:05:56.990 align:middle line:84% These are the elements in x such that x is in u1 or x 01:05:56.990 --> 01:06:01.595 align:middle line:90% is in u2 or both. 01:06:01.595 --> 01:06:04.700 align:middle line:90% 01:06:04.700 --> 01:06:09.650 align:middle line:84% And so you often illustrate this with-- 01:06:09.650 --> 01:06:10.790 align:middle line:90% this is x. 01:06:10.790 --> 01:06:13.010 align:middle line:90% This is u1. 01:06:13.010 --> 01:06:16.765 align:middle line:90% This is u2-- u1 and u2. 01:06:16.765 --> 01:06:19.370 align:middle line:90% 01:06:19.370 --> 01:06:25.430 align:middle line:84% Then this thing here is the union. 01:06:25.430 --> 01:06:29.360 align:middle line:84% And likewise, the intersection is things 01:06:29.360 --> 01:06:32.570 align:middle line:90% that is lying in both sets. 01:06:32.570 --> 01:06:44.180 align:middle line:84% So the intersection-- so this is, 01:06:44.180 --> 01:06:46.670 align:middle line:90% again, you have these two sets. 01:06:46.670 --> 01:06:49.880 align:middle line:90% And this is the intersection. 01:06:49.880 --> 01:06:55.620 align:middle line:90% And you denote it like this. 01:06:55.620 --> 01:06:58.080 align:middle line:90% And this is the x. 01:06:58.080 --> 01:07:02.350 align:middle line:90% So that x is in u1. 01:07:02.350 --> 01:07:07.650 align:middle line:90% And x is in u2. 01:07:07.650 --> 01:07:11.220 align:middle line:84% That's the intersection of these two sets. 01:07:11.220 --> 01:07:17.290 align:middle line:90% And then there is a third-- 01:07:17.290 --> 01:07:20.560 align:middle line:84% and then we talk about generalization of this. 01:07:20.560 --> 01:07:25.000 align:middle line:84% So there's a third, and this will complement. 01:07:25.000 --> 01:07:29.070 align:middle line:84% But let me also introduce the notation. 01:07:29.070 --> 01:07:38.460 align:middle line:84% So if you take this subset of x, then the complement-- and there 01:07:38.460 --> 01:07:47.410 align:middle line:84% are slightly different ways of doing it of notation for it. 01:07:47.410 --> 01:07:49.500 align:middle line:84% And I always use this one because I 01:07:49.500 --> 01:07:51.490 align:middle line:90% think that's the clearest one. 01:07:51.490 --> 01:07:55.040 align:middle line:90% So this is like x take away u. 01:07:55.040 --> 01:07:56.510 align:middle line:90% And this is totally standard. 01:07:56.510 --> 01:07:58.270 align:middle line:84% There's another one that's often used 01:07:58.270 --> 01:08:00.220 align:middle line:84% where you have C. That's slightly 01:08:00.220 --> 01:08:01.760 align:middle line:90% less standard than this one. 01:08:01.760 --> 01:08:03.920 align:middle line:84% And sometimes, it can be confusing. 01:08:03.920 --> 01:08:07.060 align:middle line:90% This one can never be confusing. 01:08:07.060 --> 01:08:09.910 align:middle line:84% And so this thing here is a compliment. 01:08:09.910 --> 01:08:14.140 align:middle line:84% And so this means that it's the x such 01:08:14.140 --> 01:08:17.069 align:middle line:90% that x does not belong to u. 01:08:17.069 --> 01:08:24.939 align:middle line:90% 01:08:24.939 --> 01:08:27.529 align:middle line:84% So that's standard operation of sets. 01:08:27.529 --> 01:08:31.029 align:middle line:84% But there is a generalization of this union 01:08:31.029 --> 01:08:33.910 align:middle line:84% here, where instead of just two sets, 01:08:33.910 --> 01:08:36.979 align:middle line:84% you could take potentially infinitely many sets. 01:08:36.979 --> 01:08:40.010 align:middle line:84% So if you take a family of subsets-- 01:08:40.010 --> 01:08:47.460 align:middle line:84% so alpha here is a family of subsets. 01:08:47.460 --> 01:08:49.979 align:middle line:90% 01:08:49.979 --> 01:08:54.130 align:middle line:84% Then you could look at the union of the u alpha. 01:08:54.130 --> 01:08:56.520 align:middle line:90% This is denoted like this. 01:08:56.520 --> 01:09:12.120 align:middle line:84% And this is the x in x such that x is in u alpha for some alpha. 01:09:12.120 --> 01:09:15.670 align:middle line:84% I mean, it could be in many of the u alphas, 01:09:15.670 --> 01:09:18.720 align:middle line:90% but it has to be at least one. 01:09:18.720 --> 01:09:21.970 align:middle line:84% So this is the union of a family. 01:09:21.970 --> 01:09:27.430 align:middle line:84% And likewise, the intersection of a family is the x. 01:09:27.430 --> 01:09:31.319 align:middle line:84% And then it has to now lie in all of them. 01:09:31.319 --> 01:09:37.790 align:middle line:84% It's x in x, where x is in u alpha for all alpha. 01:09:37.790 --> 01:09:42.210 align:middle line:90% 01:09:42.210 --> 01:09:47.500 align:middle line:84% That's a union and the intersections of family. 01:09:47.500 --> 01:10:01.750 align:middle line:84% And so now let me look at some example in particular. 01:10:01.750 --> 01:10:08.250 align:middle line:84% Let me look at an example of this. 01:10:08.250 --> 01:10:16.390 align:middle line:90% 01:10:16.390 --> 01:10:22.870 align:middle line:84% And then let me prove something about open sets. 01:10:22.870 --> 01:10:27.100 align:middle line:84% So let me look at an example here. 01:10:27.100 --> 01:10:30.730 align:middle line:90% Suppose that u alpha or u n-- 01:10:30.730 --> 01:10:32.830 align:middle line:90% I'm just going to look at-- 01:10:32.830 --> 01:10:38.540 align:middle line:84% it's going to be now a family of subsets of the real line. 01:10:38.540 --> 01:10:42.290 align:middle line:84% And they're going to be indexed by the natural number. 01:10:42.290 --> 01:10:48.710 align:middle line:84% So this thing here is going to be minus 1 over n to 1 over n. 01:10:48.710 --> 01:10:53.360 align:middle line:84% And if I'm looking at the union of these guys-- 01:10:53.360 --> 01:11:04.380 align:middle line:84% but you see, this is a sequence of open sets. 01:11:04.380 --> 01:11:07.530 align:middle line:84% I mean, it's like for each n, you have an open set. 01:11:07.530 --> 01:11:11.010 align:middle line:84% But as n gets larger, they're smaller. 01:11:11.010 --> 01:11:12.810 align:middle line:84% The first one is the biggest one. 01:11:12.810 --> 01:11:16.010 align:middle line:84% So if you take the union of these guys, 01:11:16.010 --> 01:11:20.070 align:middle line:84% that just gives you the first one you started with. 01:11:20.070 --> 01:11:23.600 align:middle line:90% So this is the union of these. 01:11:23.600 --> 01:11:26.960 align:middle line:84% If you take the intersection of them, 01:11:26.960 --> 01:11:31.820 align:middle line:84% well, the only thing that is in all of these sets, that's 01:11:31.820 --> 01:11:32.845 align:middle line:90% actually the 0. 01:11:32.845 --> 01:11:39.410 align:middle line:90% 01:11:39.410 --> 01:11:48.750 align:middle line:84% Now we need some stuff about operations of sets. 01:11:48.750 --> 01:11:52.790 align:middle line:90% 01:11:52.790 --> 01:11:56.890 align:middle line:84% And so there's some standard operations. 01:11:56.890 --> 01:12:07.000 align:middle line:84% And my recommendation is that not try to remember this. 01:12:07.000 --> 01:12:08.480 align:middle line:90% I don't remember it myself. 01:12:08.480 --> 01:12:10.400 align:middle line:90% So I always have to run through. 01:12:10.400 --> 01:12:14.270 align:middle line:84% But once you have done it once or twice, 01:12:14.270 --> 01:12:18.710 align:middle line:84% the proof is so trivial that it's not worth remembering. 01:12:18.710 --> 01:12:22.820 align:middle line:84% And I think the danger if you try to remember these things, 01:12:22.820 --> 01:12:24.620 align:middle line:90% you may remember it wrong. 01:12:24.620 --> 01:12:27.820 align:middle line:84% And that's much worse than just running through the proof. 01:12:27.820 --> 01:12:33.880 align:middle line:84% So you have the following lemma here. 01:12:33.880 --> 01:12:37.347 align:middle line:84% So you have a set, and then you have subsets. 01:12:37.347 --> 01:12:40.060 align:middle line:90% 01:12:40.060 --> 01:12:48.620 align:middle line:84% Let's call it A, B, and then a family of subsets. 01:12:48.620 --> 01:12:51.560 align:middle line:84% And the thing is the first one is 01:12:51.560 --> 01:12:56.630 align:middle line:84% that if you take the complement of the complement, 01:12:56.630 --> 01:13:01.220 align:middle line:84% so if you take A and you take everything that is not in A-- 01:13:01.220 --> 01:13:03.800 align:middle line:90% 01:13:03.800 --> 01:13:07.530 align:middle line:84% so just think about it that if you take-- so here you have x. 01:13:07.530 --> 01:13:11.120 align:middle line:84% Here, you have A. This is A. Everything that 01:13:11.120 --> 01:13:14.790 align:middle line:90% is not in A, that's over here. 01:13:14.790 --> 01:13:18.420 align:middle line:84% But then if it's not over here, then it's A. 01:13:18.420 --> 01:13:22.500 align:middle line:90% So the first one is just that. 01:13:22.500 --> 01:13:25.650 align:middle line:84% And actually, just because we're also running out of time, 01:13:25.650 --> 01:13:27.180 align:middle line:90% we really proved this. 01:13:27.180 --> 01:13:31.010 align:middle line:84% I mean, so if you have something that is not in A, 01:13:31.010 --> 01:13:32.930 align:middle line:90% then it's like here. 01:13:32.930 --> 01:13:36.030 align:middle line:84% But if it's not like here then it's like A there. 01:13:36.030 --> 01:13:38.510 align:middle line:90% That's that statement. 01:13:38.510 --> 01:13:47.500 align:middle line:84% The second one is that if you take the complement of that, 01:13:47.500 --> 01:13:51.320 align:middle line:90% it's in the union of these guys. 01:13:51.320 --> 01:13:55.830 align:middle line:90% 01:13:55.830 --> 01:14:01.020 align:middle line:84% So if it's not, this just means that it's not 01:14:01.020 --> 01:14:03.160 align:middle line:90% in any of these guys. 01:14:03.160 --> 01:14:05.890 align:middle line:90% This here is the set. 01:14:05.890 --> 01:14:09.780 align:middle line:84% Imagine that you have something that looks like-- 01:14:09.780 --> 01:14:12.930 align:middle line:84% you have a bunch of sets like this. 01:14:12.930 --> 01:14:18.310 align:middle line:84% This is supposed to illustrate the A alphas. 01:14:18.310 --> 01:14:21.420 align:middle line:84% And if it's not this, just say that if it's not 01:14:21.420 --> 01:14:29.220 align:middle line:84% in any of these, well, then the claim is that, then, it's 01:14:29.220 --> 01:14:31.470 align:middle line:90% in the intersection of-- 01:14:31.470 --> 01:14:35.640 align:middle line:84% that it's not in this one, it's not in this one, 01:14:35.640 --> 01:14:38.650 align:middle line:84% and it's not in this one, and it's not in that one. 01:14:38.650 --> 01:14:40.350 align:middle line:90% So that's also clear enough. 01:14:40.350 --> 01:14:46.280 align:middle line:90% So this is the second one. 01:14:46.280 --> 01:14:52.820 align:middle line:84% Again, I actually wrote out the proof in the notes. 01:14:52.820 --> 01:14:54.715 align:middle line:90% So you can check that out. 01:14:54.715 --> 01:15:03.500 align:middle line:90% 01:15:03.500 --> 01:15:16.010 align:middle line:84% And the third operation is that if you take x, 01:15:16.010 --> 01:15:22.060 align:middle line:84% and you are removing the intersection of this family-- 01:15:22.060 --> 01:15:24.800 align:middle line:90% 01:15:24.800 --> 01:15:27.990 align:middle line:90% so this means that it is not-- 01:15:27.990 --> 01:15:30.470 align:middle line:84% so this means that the intersection 01:15:30.470 --> 01:15:35.490 align:middle line:84% of the family in this picture here would be this thing here. 01:15:35.490 --> 01:15:38.710 align:middle line:90% And so it's not right there. 01:15:38.710 --> 01:15:41.010 align:middle line:90% Everything else is allowed. 01:15:41.010 --> 01:15:42.520 align:middle line:90% But that's the same. 01:15:42.520 --> 01:15:47.990 align:middle line:84% You see, that's the same as the union of x take away 01:15:47.990 --> 01:15:58.680 align:middle line:84% A. This here is the element that is not in one of-- there's 01:15:58.680 --> 01:16:01.620 align:middle line:84% at least one of the u alpha, A alpha, 01:16:01.620 --> 01:16:05.730 align:middle line:84% where it's not in, and so that's the same. 01:16:05.730 --> 01:16:09.780 align:middle line:90% So this is the same as this. 01:16:09.780 --> 01:16:14.470 align:middle line:84% Again, I wrote it out in the notes-- 01:16:14.470 --> 01:16:16.690 align:middle line:84% the proof, actually, of these things. 01:16:16.690 --> 01:16:20.940 align:middle line:90% 01:16:20.940 --> 01:16:23.190 align:middle line:90% And let me just say-- 01:16:23.190 --> 01:16:27.480 align:middle line:84% maybe we can't quite I mean, we'll pick up in the notes 01:16:27.480 --> 01:16:30.790 align:middle line:90% where the other stuff-- 01:16:30.790 --> 01:16:36.410 align:middle line:84% but let me just say that when you prove of thing, 01:16:36.410 --> 01:16:43.600 align:middle line:84% and we even did this here when I'm just illustrating it 01:16:43.600 --> 01:16:45.040 align:middle line:90% with a picture. 01:16:45.040 --> 01:16:51.650 align:middle line:84% So we prove it using the following trivial lemma. 01:16:51.650 --> 01:16:54.610 align:middle line:84% And I wrote that out as a lemma in the notes, 01:16:54.610 --> 01:16:59.180 align:middle line:84% that if you are trying to prove that two sets are equal-- 01:16:59.180 --> 01:17:02.500 align:middle line:84% so suppose you're trying to prove that two sets are equal. 01:17:02.500 --> 01:17:07.070 align:middle line:84% Then, in a way, it's natural to divide this into two parts. 01:17:07.070 --> 01:17:23.950 align:middle line:84% So this is the same as proving that A is contained in B and B 01:17:23.950 --> 01:17:27.760 align:middle line:84% is contained in A. So if you're trying 01:17:27.760 --> 01:17:29.810 align:middle line:84% to prove that two sets are equal, 01:17:29.810 --> 01:17:34.860 align:middle line:84% then you would prove that for any elements in A, it's also B, 01:17:34.860 --> 01:17:36.120 align:middle line:90% this is this thing. 01:17:36.120 --> 01:17:39.940 align:middle line:84% And you prove that for any element B, it's also in A. 01:17:39.940 --> 01:17:46.690 align:middle line:84% And so when I wrote out the proof of these three things, 01:17:46.690 --> 01:17:51.030 align:middle line:84% then I wrote it out like each of them has two parts that you 01:17:51.030 --> 01:17:53.385 align:middle line:84% prove that this set is contained in that 01:17:53.385 --> 01:17:55.260 align:middle line:84% and you prove that this is contained in that. 01:17:55.260 --> 01:17:59.580 align:middle line:90% 01:17:59.580 --> 01:18:05.370 align:middle line:90% Now we're almost actually there. 01:18:05.370 --> 01:18:15.630 align:middle line:84% So now let me just say that-- so that's actually 01:18:15.630 --> 01:18:21.430 align:middle line:90% in one of the Pset problems. 01:18:21.430 --> 01:18:24.450 align:middle line:90% But there's no Pset. 01:18:24.450 --> 01:18:27.480 align:middle line:84% You have a little break now for Psets 01:18:27.480 --> 01:18:31.020 align:middle line:84% because I want to make sure that there is no Pset material we 01:18:31.020 --> 01:18:34.330 align:middle line:84% haven't yet covered in class, so that's why there's a break. 01:18:34.330 --> 01:18:39.020 align:middle line:84% And anyway, all of the Psets at this point are actually posted. 01:18:39.020 --> 01:18:41.330 align:middle line:90% So the last one I think is due-- 01:18:41.330 --> 01:18:44.330 align:middle line:84% I forget whether it's the 8 of May or something like that. 01:18:44.330 --> 01:18:46.790 align:middle line:90% And again, there's five left. 01:18:46.790 --> 01:18:48.940 align:middle line:90% You've already done [INAUDIBLE]. 01:18:48.940 --> 01:18:57.650 align:middle line:84% So now if x here, d is a metric space, 01:18:57.650 --> 01:19:00.490 align:middle line:84% then you have the following lemma 01:19:00.490 --> 01:19:09.670 align:middle line:84% that if O alpha are open subsets, 01:19:09.670 --> 01:19:13.890 align:middle line:84% then the union of the O alpha is also-- 01:19:13.890 --> 01:19:20.200 align:middle line:90% 01:19:20.200 --> 01:19:22.340 align:middle line:84% But this is quite trivial, as you see. 01:19:22.340 --> 01:19:23.810 align:middle line:90% I'll just indicate why. 01:19:23.810 --> 01:19:26.150 align:middle line:84% But it's actually left on a Pset. 01:19:26.150 --> 01:19:27.320 align:middle line:90% So the proof here-- 01:19:27.320 --> 01:19:30.320 align:middle line:90% 01:19:30.320 --> 01:19:32.600 align:middle line:84% but I will just illustrate it anyway 01:19:32.600 --> 01:19:40.340 align:middle line:84% that if you take an element here in the union, well, 01:19:40.340 --> 01:19:44.540 align:middle line:84% this means that there exists a value added. 01:19:44.540 --> 01:19:50.120 align:middle line:84% So it means that x is in O alpha for some alpha. 01:19:50.120 --> 01:19:56.030 align:middle line:90% 01:19:56.030 --> 01:19:59.480 align:middle line:84% But this means that there is a ball around it 01:19:59.480 --> 01:20:01.140 align:middle line:90% because this here is open. 01:20:01.140 --> 01:20:03.890 align:middle line:84% It means that there's a little ball around it 01:20:03.890 --> 01:20:08.480 align:middle line:90% that is contained in this. 01:20:08.480 --> 01:20:11.390 align:middle line:84% But therefore, it's also contained 01:20:11.390 --> 01:20:17.270 align:middle line:90% in the union of these guys. 01:20:17.270 --> 01:20:20.270 align:middle line:84% And so this is really just a proof 01:20:20.270 --> 01:20:24.580 align:middle line:84% that the union of open sets are open. 01:20:24.580 --> 01:20:27.138 align:middle line:90% 01:20:27.138 --> 01:20:28.930 align:middle line:84% We're out of time now, but next time, we'll 01:20:28.930 --> 01:20:32.170 align:middle line:84% talk about why if you take finitely many 01:20:32.170 --> 01:20:39.170 align:middle line:84% sets, why the intersection of finitely many open sets, 01:20:39.170 --> 01:20:40.960 align:middle line:90% why that is also open. 01:20:40.960 --> 01:20:44.420 align:middle line:84% But for intersection of open sets, 01:20:44.420 --> 01:20:48.760 align:middle line:84% you have to be careful because for union, you 01:20:48.760 --> 01:20:51.860 align:middle line:84% can take the union of any number of open sets. 01:20:51.860 --> 01:20:53.000 align:middle line:90% It's always open. 01:20:53.000 --> 01:20:58.280 align:middle line:84% But if you take intersection u, in general, 01:20:58.280 --> 01:21:02.300 align:middle line:84% it's only true that finite intersections are open. 01:21:02.300 --> 01:21:04.390 align:middle line:84% And the example is one of the examples 01:21:04.390 --> 01:21:09.950 align:middle line:84% we looked at before where you take these sets here. 01:21:09.950 --> 01:21:13.030 align:middle line:90% These sets are open subsets. 01:21:13.030 --> 01:21:16.130 align:middle line:84% But if you take the intersection of all of them, 01:21:16.130 --> 01:21:17.980 align:middle line:90% then you just get 0. 01:21:17.980 --> 01:21:20.530 align:middle line:90% And that's not an open subset. 01:21:20.530 --> 01:21:24.720 align:middle line:90% So we're done. 01:21:24.720 --> 01:21:27.000 align:middle line:90%