WEBVTT 00:00:00.000 --> 00:00:01.952 align:middle line:90% [SQUEAKING] 00:00:01.952 --> 00:00:03.416 align:middle line:90% [RUSTLING] 00:00:03.416 --> 00:00:04.880 align:middle line:90% [CLICKING] 00:00:04.880 --> 00:00:10.633 align:middle line:90% 00:00:10.633 --> 00:00:12.050 align:middle line:84% PAIGE BRIGHT: Last time we started 00:00:12.050 --> 00:00:15.410 align:middle line:84% talking about metric spaces being compact on Euclidean 00:00:15.410 --> 00:00:17.720 align:middle line:84% space in particular, where we showed 00:00:17.720 --> 00:00:21.860 align:middle line:84% on Rn that sequentially compact was the same as 00:00:21.860 --> 00:00:23.900 align:middle line:84% closed and bounded, which was the same as 00:00:23.900 --> 00:00:25.610 align:middle line:90% topologically compact. 00:00:25.610 --> 00:00:27.560 align:middle line:84% And notice that in these two theorems, 00:00:27.560 --> 00:00:30.020 align:middle line:84% we use some very important propositions that we've 00:00:30.020 --> 00:00:32.150 align:middle line:90% learned from real analysis. 00:00:32.150 --> 00:00:34.910 align:middle line:84% To prove that closed and bounded implied sequentially compact, 00:00:34.910 --> 00:00:37.920 align:middle line:84% we have to use the Bolzano-Weierstrass theorem, 00:00:37.920 --> 00:00:40.250 align:middle line:84% which essentially states exactly the condition 00:00:40.250 --> 00:00:43.160 align:middle line:90% of sequentially compactness. 00:00:43.160 --> 00:00:44.415 align:middle line:90% Bolzano-Weierstrass. 00:00:44.415 --> 00:00:48.130 align:middle line:90% 00:00:48.130 --> 00:00:51.490 align:middle line:84% And to use closed and bounded, to show topologically compact, 00:00:51.490 --> 00:00:56.960 align:middle line:84% we had to use the Heine-Borel theorem, 00:00:56.960 --> 00:01:00.620 align:middle line:84% where we trapped our compact set inside of a closed cube. 00:01:00.620 --> 00:01:04.849 align:middle line:84% And we know that that closed cube is going to be compact, 00:01:04.849 --> 00:01:08.070 align:middle line:84% but we don't always have these closed cubes on metric spaces, 00:01:08.070 --> 00:01:08.570 align:middle line:90% right? 00:01:08.570 --> 00:01:13.010 align:middle line:84% What does it mean for there to be a cube of functions? 00:01:13.010 --> 00:01:15.110 align:middle line:84% And in fact, in general, these implications 00:01:15.110 --> 00:01:17.150 align:middle line:84% are not going to be precisely true. 00:01:17.150 --> 00:01:19.670 align:middle line:84% And small error from last time-- last time, 00:01:19.670 --> 00:01:22.370 align:middle line:84% I stated that sequentially compact was the same as closed 00:01:22.370 --> 00:01:23.690 align:middle line:90% and bounded on metric spaces. 00:01:23.690 --> 00:01:26.510 align:middle line:90% This is not the case. 00:01:26.510 --> 00:01:28.760 align:middle line:90% Sorry about that. 00:01:28.760 --> 00:01:32.640 align:middle line:84% We are using Bolzano-Weierstrass to prove this theorem. 00:01:32.640 --> 00:01:35.070 align:middle line:84% And we need it to show the other implication. 00:01:35.070 --> 00:01:37.410 align:middle line:84% If it was always the same, then all three of these 00:01:37.410 --> 00:01:38.430 align:middle line:90% would always be true. 00:01:38.430 --> 00:01:39.240 align:middle line:90% And it's not. 00:01:39.240 --> 00:01:41.070 align:middle line:84% Metric spaces where closed and bounded 00:01:41.070 --> 00:01:44.640 align:middle line:84% implies topologically compact have the Heine-Borel property. 00:01:44.640 --> 00:01:47.740 align:middle line:84% And not every metric space has this property. 00:01:47.740 --> 00:01:50.040 align:middle line:84% But what we're going to show today 00:01:50.040 --> 00:01:52.050 align:middle line:90% is that if we're not on Rn-- 00:01:52.050 --> 00:01:56.310 align:middle line:84% if we're on a general metric space, 00:01:56.310 --> 00:01:59.205 align:middle line:84% we still have the following implications. 00:01:59.205 --> 00:02:01.080 align:middle line:84% We already know that sequentially compact and 00:02:01.080 --> 00:02:04.200 align:middle line:84% topologically compact implies closed and bounded, which 00:02:04.200 --> 00:02:05.820 align:middle line:90% is a very helpful thing, right? 00:02:05.820 --> 00:02:08.893 align:middle line:84% If you want to show that a metric space isn't compact, 00:02:08.893 --> 00:02:11.310 align:middle line:84% you just have to show that it's not closed or at least not 00:02:11.310 --> 00:02:12.570 align:middle line:90% bounded. 00:02:12.570 --> 00:02:15.180 align:middle line:84% But we're still going to have the following implication-- 00:02:15.180 --> 00:02:16.560 align:middle line:90% what we're going to show today. 00:02:16.560 --> 00:02:19.560 align:middle line:84% We're going to show that sequentially compact is 00:02:19.560 --> 00:02:21.060 align:middle line:84% the same as topologically compact. 00:02:21.060 --> 00:02:24.240 align:middle line:90% 00:02:24.240 --> 00:02:26.580 align:middle line:84% In fact, we're going to show two more properties that 00:02:26.580 --> 00:02:29.280 align:middle line:84% emphasize the importance of compact metric spaces 00:02:29.280 --> 00:02:30.210 align:middle line:90% in general. 00:02:30.210 --> 00:02:32.190 align:middle line:84% It's going to be a very proof-intensive day, 00:02:32.190 --> 00:02:38.670 align:middle line:84% but I hope you stick with it with me. 00:02:38.670 --> 00:02:41.010 align:middle line:84% So yeah, that's our goal for today. 00:02:41.010 --> 00:02:43.440 align:middle line:84% Before we show sequentially compact implies 00:02:43.440 --> 00:02:46.050 align:middle line:84% topologically compact, and furthermore, 00:02:46.050 --> 00:02:47.880 align:middle line:84% the opposite direction, first, we're 00:02:47.880 --> 00:02:49.740 align:middle line:84% going to start by proving some lemmas 00:02:49.740 --> 00:02:51.660 align:middle line:84% about sequential compactness to make our job 00:02:51.660 --> 00:02:53.800 align:middle line:84% just a little bit easier in the long term. 00:02:53.800 --> 00:02:55.290 align:middle line:84% And in fact, these lemmas are going 00:02:55.290 --> 00:02:57.190 align:middle line:84% to be used throughout the rest of the class, 00:02:57.190 --> 00:03:00.000 align:middle line:90% so please pay attention. 00:03:00.000 --> 00:03:03.060 align:middle line:84% So first, we have what is known as the lemma-- 00:03:03.060 --> 00:03:04.425 align:middle line:90% the big number lemma-- 00:03:04.425 --> 00:03:11.820 align:middle line:90% 00:03:11.820 --> 00:03:13.650 align:middle line:84% where here I'm just not writing out number. 00:03:13.650 --> 00:03:16.210 align:middle line:84% But the idea is given our metric space 00:03:16.210 --> 00:03:25.095 align:middle line:84% x is sequentially compact, if I have an open cover of x-- 00:03:25.095 --> 00:03:29.260 align:middle line:90% 00:03:29.260 --> 00:03:30.895 align:middle line:90% let's call this our Ui's-- 00:03:30.895 --> 00:03:34.160 align:middle line:90% 00:03:34.160 --> 00:03:38.750 align:middle line:84% then there exists an R bigger than 0 such 00:03:38.750 --> 00:03:43.700 align:middle line:84% that for all x in x, the ball of radius r around 00:03:43.700 --> 00:03:51.020 align:middle line:84% x is contained in one of the Ui's for some i. 00:03:51.020 --> 00:03:53.210 align:middle line:84% Now, let's draw a picture of what I'm actually 00:03:53.210 --> 00:03:55.700 align:middle line:84% describing here because this quantifier 00:03:55.700 --> 00:03:57.200 align:middle line:84% logic can be a little bit confusing 00:03:57.200 --> 00:03:58.980 align:middle line:90% the first time you see it. 00:03:58.980 --> 00:03:59.720 align:middle line:90% So here. 00:03:59.720 --> 00:04:04.947 align:middle line:84% Let's just draw a little blob for a metric space x. 00:04:04.947 --> 00:04:07.280 align:middle line:84% And what we're going to do is find an open cover of this 00:04:07.280 --> 00:04:07.940 align:middle line:90% by the Ui. 00:04:07.940 --> 00:04:09.200 align:middle line:90% So I'll draw these by circles. 00:04:09.200 --> 00:04:12.590 align:middle line:90% 00:04:12.590 --> 00:04:13.780 align:middle line:90% So these are our Ui's. 00:04:13.780 --> 00:04:20.760 align:middle line:90% 00:04:20.760 --> 00:04:22.089 align:middle line:90% OK, there we go. 00:04:22.089 --> 00:04:25.230 align:middle line:84% And now, for every x in x, we claim 00:04:25.230 --> 00:04:28.170 align:middle line:84% that there exists a ball of radius R 00:04:28.170 --> 00:04:31.470 align:middle line:84% around x such that it's contained in only 00:04:31.470 --> 00:04:35.520 align:middle line:90% one of these open sets-- 00:04:35.520 --> 00:04:38.430 align:middle line:84% or sorry, not only one-- that it's fully 00:04:38.430 --> 00:04:40.050 align:middle line:90% contained in one of these sets. 00:04:40.050 --> 00:04:43.940 align:middle line:90% 00:04:43.940 --> 00:04:46.670 align:middle line:84% And this might ring out to you as being 00:04:46.670 --> 00:04:49.010 align:middle line:84% true if you want to prove topologically compact 00:04:49.010 --> 00:04:50.805 align:middle line:84% is the same as sequentially compact. 00:04:50.805 --> 00:04:52.430 align:middle line:84% What we're going to do in the long term 00:04:52.430 --> 00:04:54.560 align:middle line:84% is assume that we're sequentially compact, 00:04:54.560 --> 00:05:00.410 align:middle line:84% take an open cover, and use this R to construct our conclusion. 00:05:00.410 --> 00:05:02.810 align:middle line:84% But first, we have to prove this fact. 00:05:02.810 --> 00:05:06.205 align:middle line:84% And the proof is going to be pretty 00:05:06.205 --> 00:05:07.580 align:middle line:84% usual for sequential compactness. 00:05:07.580 --> 00:05:11.540 align:middle line:84% So our proof is going to be by contradiction. 00:05:11.540 --> 00:05:20.023 align:middle line:84% We're going to use contradiction, assume not true. 00:05:20.023 --> 00:05:22.190 align:middle line:84% What does it mean for this statement to be not true? 00:05:22.190 --> 00:05:26.390 align:middle line:84% Well, it means that for every R bigger than 0, 00:05:26.390 --> 00:05:34.000 align:middle line:84% there exists some x in x such that the ball of radius R 00:05:34.000 --> 00:05:39.070 align:middle line:84% around an x is not contained in any of the Ui's. 00:05:39.070 --> 00:05:40.900 align:middle line:84% It's contained in a cover of the Ui's. 00:05:40.900 --> 00:05:45.105 align:middle line:84% It's just not fully contained in one of them, at least. 00:05:45.105 --> 00:05:46.480 align:middle line:84% And what we're going to do is use 00:05:46.480 --> 00:05:49.510 align:middle line:84% this to construct a sequence and take the convergent subsequence 00:05:49.510 --> 00:05:51.230 align:middle line:90% and reach contradiction. 00:05:51.230 --> 00:06:00.580 align:middle line:84% So for each R equal to 1 over n, choose 00:06:00.580 --> 00:06:07.310 align:middle line:84% x n to satisfy the above property. 00:06:07.310 --> 00:06:09.710 align:middle line:84% So we're going to choose the points in our sequence to be 00:06:09.710 --> 00:06:12.410 align:middle line:84% the xn, or at least one of the xn, 00:06:12.410 --> 00:06:15.920 align:middle line:84% such that the ball of radius R or the ball of radius 1 over n 00:06:15.920 --> 00:06:18.527 align:middle line:84% around x is not contained in any of the Ui's. 00:06:18.527 --> 00:06:21.110 align:middle line:84% And we know that this is going to be true because one of our n 00:06:21.110 --> 00:06:23.960 align:middle line:90% is going to be bigger than 0. 00:06:23.960 --> 00:06:25.880 align:middle line:84% So what are we going to do with the sequence? 00:06:25.880 --> 00:06:27.560 align:middle line:84% Well, we're going to start by taking 00:06:27.560 --> 00:06:29.120 align:middle line:90% a convergent subsequence, right? 00:06:29.120 --> 00:06:33.560 align:middle line:84% We're going to use the fact that we're sequentially compact. 00:06:33.560 --> 00:06:35.820 align:middle line:84% And this trend will carry on throughout the day. 00:06:35.820 --> 00:06:42.170 align:middle line:84% So we have a convergent subsequence-- 00:06:42.170 --> 00:06:46.040 align:middle line:90% 00:06:46.040 --> 00:06:47.690 align:middle line:90% let's call it xnk-- 00:06:47.690 --> 00:06:49.760 align:middle line:90% converging to x, of course. 00:06:49.760 --> 00:06:52.220 align:middle line:84% That's what we usually let it be. 00:06:52.220 --> 00:06:52.910 align:middle line:90% OK. 00:06:52.910 --> 00:06:54.260 align:middle line:90% So what are we going to do? 00:06:54.260 --> 00:06:55.820 align:middle line:84% We want to reach the contradiction 00:06:55.820 --> 00:06:59.180 align:middle line:84% that there is, in fact, a ball of radius R around one 00:06:59.180 --> 00:07:01.610 align:middle line:84% of these x's contained in the Ui. 00:07:01.610 --> 00:07:02.550 align:middle line:90% How do we do that? 00:07:02.550 --> 00:07:08.030 align:middle line:84% Well first, we notice that x has to be in one of the Ui's. 00:07:08.030 --> 00:07:11.970 align:middle line:90% x in Ui0 for some i0. 00:07:11.970 --> 00:07:16.700 align:middle line:84% And this is true given that it's an open cover of our space. 00:07:16.700 --> 00:07:21.290 align:middle line:84% And therefore, because the Ui0 is open, 00:07:21.290 --> 00:07:28.070 align:middle line:84% there exists an R0 bigger than 0 such that the ball of radius R0 00:07:28.070 --> 00:07:29.420 align:middle line:90% is contained in Ui0. 00:07:29.420 --> 00:07:32.450 align:middle line:90% 00:07:32.450 --> 00:07:32.950 align:middle line:90% Right? 00:07:32.950 --> 00:07:36.820 align:middle line:84% Now, we want to find a subset of this such 00:07:36.820 --> 00:07:38.330 align:middle line:90% that we reach for contradiction. 00:07:38.330 --> 00:07:56.540 align:middle line:84% So choose n sufficiently large so that 1 over n 00:07:56.540 --> 00:08:08.120 align:middle line:84% is less than R0 over 2 and do the distance from x to xn 00:08:08.120 --> 00:08:10.670 align:middle line:90% is less than R0 over 2. 00:08:10.670 --> 00:08:15.560 align:middle line:84% And we're going to use this to create a subset of b R0 of x. 00:08:15.560 --> 00:08:16.570 align:middle line:90% So how do we do that? 00:08:16.570 --> 00:08:27.550 align:middle line:84% Well, we consider the ball of radius 1 over n around x-- 00:08:27.550 --> 00:08:28.720 align:middle line:90% or sorry, around xn. 00:08:28.720 --> 00:08:31.730 align:middle line:90% 00:08:31.730 --> 00:08:36.409 align:middle line:84% So for all y in this set, we notice 00:08:36.409 --> 00:08:46.220 align:middle line:84% that the distance from x to y is less than 00:08:46.220 --> 00:08:52.430 align:middle line:84% or equal to the distance from x to xn plus the distance from y 00:08:52.430 --> 00:08:54.320 align:middle line:90% to xn. 00:08:54.320 --> 00:08:56.520 align:middle line:84% This is by the triangle inequality. 00:08:56.520 --> 00:09:03.350 align:middle line:84% This term is going to be less than R0 over 2. 00:09:03.350 --> 00:09:07.330 align:middle line:84% It's less than 1 over n, which is less than R0 over 2 00:09:07.330 --> 00:09:10.990 align:middle line:84% plus 1 over something that's less than 1 00:09:10.990 --> 00:09:15.030 align:middle line:84% over n, which is less than R0 over 2. 00:09:15.030 --> 00:09:18.450 align:middle line:84% So this is R0, which tells us that the ball of radius 00:09:18.450 --> 00:09:26.680 align:middle line:84% 1 over n around x is fully contained. 00:09:26.680 --> 00:09:32.980 align:middle line:84% And the ball of radius R0 around x, which is fully contained, 00:09:32.980 --> 00:09:36.040 align:middle line:84% and Ui0, which is our contradiction, right? 00:09:36.040 --> 00:09:38.980 align:middle line:84% Because we claimed at the very beginning that none of these 00:09:38.980 --> 00:09:44.710 align:middle line:84% balls are going to be fully contained in any of the Ui's. 00:09:44.710 --> 00:09:47.170 align:middle line:90% So this is the big number lemma. 00:09:47.170 --> 00:09:49.540 align:middle line:84% As a small note, the largest such R 00:09:49.540 --> 00:09:52.660 align:middle line:84% such that this property holds is known as the Lebesgue number. 00:09:52.660 --> 00:09:55.233 align:middle line:84% And this comes up in a number of other applications. 00:09:55.233 --> 00:09:57.400 align:middle line:84% We're just not going to be talking about them today. 00:09:57.400 --> 00:10:04.240 align:middle line:90% 00:10:04.240 --> 00:10:09.450 align:middle line:84% So now, we have a way of refining our open cover 00:10:09.450 --> 00:10:12.510 align:middle line:84% to refine our open cover to the set of open balls of all 00:10:12.510 --> 00:10:15.330 align:middle line:84% the same radius, but what we need to do 00:10:15.330 --> 00:10:23.590 align:middle line:84% is get finitely many of these open balls for our open cover. 00:10:23.590 --> 00:10:28.720 align:middle line:84% And as such, we're going to define a new term. 00:10:28.720 --> 00:10:34.310 align:middle line:84% So definition-- the term is known as totally bounded. 00:10:34.310 --> 00:10:39.180 align:middle line:90% 00:10:39.180 --> 00:10:41.580 align:middle line:84% A-- or I should say metric space-- 00:10:41.580 --> 00:10:45.660 align:middle line:90% 00:10:45.660 --> 00:10:47.370 align:middle line:90% let's call it x-- 00:10:47.370 --> 00:10:58.990 align:middle line:84% is totally bounded if, for all epsilon bigger than 0, 00:10:58.990 --> 00:11:05.980 align:middle line:84% there exists y1 through yk such that the union of balls 00:11:05.980 --> 00:11:14.240 align:middle line:84% of radius epsilon around the finitely many yi's contains x. 00:11:14.240 --> 00:11:16.160 align:middle line:84% So it's a very similar picture to this, 00:11:16.160 --> 00:11:19.370 align:middle line:84% only we have all the same radius R instead 00:11:19.370 --> 00:11:22.220 align:middle line:90% of our regular open cover. 00:11:22.220 --> 00:11:23.720 align:middle line:84% And what we're going to do is show 00:11:23.720 --> 00:11:27.290 align:middle line:84% that sequential compactness implies totally bounded. 00:11:27.290 --> 00:11:30.700 align:middle line:90% This is another lemma of ours. 00:11:30.700 --> 00:11:31.880 align:middle line:90% OK. 00:11:31.880 --> 00:11:46.000 align:middle line:84% Lemma-- sequentially compact implies totally 00:11:46.000 --> 00:11:50.460 align:middle line:90% bounded as metric spaces. 00:11:50.460 --> 00:11:52.760 align:middle line:84% So if your metric space is sequentially compact, 00:11:52.760 --> 00:11:55.010 align:middle line:84% then it's going to be totally bounded. 00:11:55.010 --> 00:11:55.880 align:middle line:90% Let's prove this. 00:11:55.880 --> 00:12:00.710 align:middle line:84% Proof-- again, our proof is going to be by contradiction. 00:12:00.710 --> 00:12:03.410 align:middle line:90% 00:12:03.410 --> 00:12:05.540 align:middle line:84% Assume that it's not totally bounded. 00:12:05.540 --> 00:12:10.700 align:middle line:84% Then for all epsilon bigger than 0, there exists-- 00:12:10.700 --> 00:12:18.160 align:middle line:84% or sorry, I should say there does not 00:12:18.160 --> 00:12:29.195 align:middle line:84% exist finitely many epsilon balls that cover our space. 00:12:29.195 --> 00:12:35.470 align:middle line:90% 00:12:35.470 --> 00:12:37.617 align:middle line:84% What does it mean to be not existent, right? 00:12:37.617 --> 00:12:40.200 align:middle line:84% What does it mean for there to not exist finitely many epsilon 00:12:40.200 --> 00:12:40.830 align:middle line:90% balls? 00:12:40.830 --> 00:12:43.320 align:middle line:84% Well, this means that there are infinitely many of them. 00:12:43.320 --> 00:12:47.150 align:middle line:90% 00:12:47.150 --> 00:12:54.375 align:middle line:84% So it takes epsilon balls to cover x. 00:12:54.375 --> 00:12:57.305 align:middle line:90% 00:12:57.305 --> 00:12:58.930 align:middle line:84% Now, what are we going to do with this? 00:12:58.930 --> 00:13:02.410 align:middle line:84% We're once again going to construct a sequence 00:13:02.410 --> 00:13:04.150 align:middle line:90% and a convergent subsequence. 00:13:04.150 --> 00:13:06.520 align:middle line:84% But we need to somehow use the fact 00:13:06.520 --> 00:13:10.900 align:middle line:84% that it takes infinitely many epsilon balls to cover x. 00:13:10.900 --> 00:13:14.110 align:middle line:84% Well, let's just start by taking any x-- 00:13:14.110 --> 00:13:18.130 align:middle line:90% x1-- be an x. 00:13:18.130 --> 00:13:23.440 align:middle line:84% And now we want to construct an x2 such that there is no Cauchy 00:13:23.440 --> 00:13:25.400 align:middle line:90% subsequence of the xn's. 00:13:25.400 --> 00:13:31.090 align:middle line:84% So let's take x2 to be an x not including 00:13:31.090 --> 00:13:34.310 align:middle line:84% the ball of radius epsilon around x1. 00:13:34.310 --> 00:13:37.520 align:middle line:90% Let's continue this process. 00:13:37.520 --> 00:13:43.100 align:middle line:84% Let xn be in x not including the union of balls 00:13:43.100 --> 00:13:44.930 align:middle line:90% of radius epsilon around xi. 00:13:44.930 --> 00:13:47.640 align:middle line:90% 00:13:47.640 --> 00:13:49.390 align:middle line:90% Now, why are we doing this? 00:13:49.390 --> 00:13:51.540 align:middle line:84% We want to create a sequence of points 00:13:51.540 --> 00:13:54.240 align:middle line:84% such that the distance between every point 00:13:54.240 --> 00:13:58.080 align:middle line:84% is bigger than epsilon, which we have immediately from this. 00:13:58.080 --> 00:14:03.780 align:middle line:84% The distance from xi to xj is always bigger than epsilon 00:14:03.780 --> 00:14:06.600 align:middle line:84% by construction because if it wasn't less than epsilon, 00:14:06.600 --> 00:14:09.270 align:middle line:84% then it would be contained in the ball of radius epsilon 00:14:09.270 --> 00:14:12.860 align:middle line:90% around one of the other xi's. 00:14:12.860 --> 00:14:15.060 align:middle line:90% So what does this tell us? 00:14:15.060 --> 00:14:19.490 align:middle line:84% This tells us that our sequence can never be Cauchy, right? 00:14:19.490 --> 00:14:23.510 align:middle line:84% If our sequence was Cauchy, or a subsequence was Cauchy, 00:14:23.510 --> 00:14:26.090 align:middle line:84% then the distance between subsequent points 00:14:26.090 --> 00:14:32.570 align:middle line:84% has to be able to get less than epsilon, which is not the case. 00:14:32.570 --> 00:14:48.040 align:middle line:84% So what we notice is that there are no Cauchy subsequences 00:14:48.040 --> 00:14:49.150 align:middle line:90% of the sequence xi. 00:14:49.150 --> 00:14:51.950 align:middle line:90% 00:14:51.950 --> 00:14:53.930 align:middle line:84% And this is going to be our contradiction. 00:14:53.930 --> 00:14:54.980 align:middle line:90% Why? 00:14:54.980 --> 00:14:57.950 align:middle line:84% Because this is a sequence in a sequentially compact metric 00:14:57.950 --> 00:14:58.560 align:middle line:90% space. 00:14:58.560 --> 00:15:02.510 align:middle line:84% So we know that there has to exist a convergent subsequence. 00:15:02.510 --> 00:15:05.570 align:middle line:84% But being convergent implies that it's going to be Cauchy. 00:15:05.570 --> 00:15:07.190 align:middle line:84% We've shown this on the second day 00:15:07.190 --> 00:15:10.350 align:middle line:84% in our general theory of metric spaces. 00:15:10.350 --> 00:15:12.740 align:middle line:84% So what we've shown is that there 00:15:12.740 --> 00:15:14.780 align:middle line:84% should be a Cauchy subsequence, but there 00:15:14.780 --> 00:15:16.850 align:middle line:90% is no Cauchy subsequence. 00:15:16.850 --> 00:15:19.170 align:middle line:90% So this is our contradiction. 00:15:19.170 --> 00:15:22.790 align:middle line:90% And that concludes the proof. 00:15:22.790 --> 00:15:24.280 align:middle line:90% And these two lemmas-- 00:15:24.280 --> 00:15:27.100 align:middle line:84% the Lebesgue number lemma and this one right here-- 00:15:27.100 --> 00:15:29.830 align:middle line:84% are going to be all we need to show that sequentially compact 00:15:29.830 --> 00:15:31.900 align:middle line:90% implies topologically compact. 00:15:31.900 --> 00:15:33.610 align:middle line:90% So let's start that proof. 00:15:33.610 --> 00:15:48.060 align:middle line:84% Theorem-- sequentially compact is the same as topologically 00:15:48.060 --> 00:15:49.575 align:middle line:90% compact as metric spaces. 00:15:49.575 --> 00:15:56.380 align:middle line:90% 00:15:56.380 --> 00:15:58.360 align:middle line:90% So let's start the proof. 00:15:58.360 --> 00:16:10.050 align:middle line:84% Proof-- assume sequentially compact, 00:16:10.050 --> 00:16:18.400 align:middle line:84% and let Ui be an open cover of the metric space x. 00:16:18.400 --> 00:16:23.010 align:middle line:90% 00:16:23.010 --> 00:16:26.100 align:middle line:84% What we then know by the Lebesgue number lemma is that 00:16:26.100 --> 00:16:29.820 align:middle line:84% there exists an R bigger than 0 such that balls of radius R are 00:16:29.820 --> 00:16:32.010 align:middle line:90% contained in one of the Ui's. 00:16:32.010 --> 00:16:40.490 align:middle line:84% So therefore, by the Lebesgue number lemma, 00:16:40.490 --> 00:16:45.160 align:middle line:84% there exists an R bigger than 0 such that balls 00:16:45.160 --> 00:16:50.270 align:middle line:84% of radius all around x are contained in Ui for some i. 00:16:50.270 --> 00:16:53.620 align:middle line:90% 00:16:53.620 --> 00:16:58.150 align:middle line:84% I.e. for every x, there exists some i such that this is true. 00:16:58.150 --> 00:17:02.920 align:middle line:84% And now, how do we go from this to finitely many of these Ui's? 00:17:02.920 --> 00:17:04.869 align:middle line:84% Well, here we use the following lemma. 00:17:04.869 --> 00:17:06.680 align:middle line:84% We know that it's totally bounded. 00:17:06.680 --> 00:17:10.660 align:middle line:84% So given this R bigger than 0, let R 00:17:10.660 --> 00:17:15.609 align:middle line:84% be equal to epsilon and the definition of totally bounded. 00:17:15.609 --> 00:17:27.750 align:middle line:84% And then we know that there exists finitely many y1 00:17:27.750 --> 00:17:36.700 align:middle line:84% through yk such that x is contained in balls of radius 00:17:36.700 --> 00:17:41.860 align:middle line:84% in the union of balls of radius R around yi 00:17:41.860 --> 00:17:45.010 align:middle line:90% from i equals 1 to k. 00:17:45.010 --> 00:17:51.760 align:middle line:84% But we want to go from this to a finite subcover. 00:17:51.760 --> 00:17:55.690 align:middle line:84% But what we notice is that each of these balls of radius R is 00:17:55.690 --> 00:17:58.780 align:middle line:90% contained in one of the Ui's. 00:17:58.780 --> 00:18:02.380 align:middle line:84% So right now, this is going to be contained 00:18:02.380 --> 00:18:06.700 align:middle line:84% in the union of Uij, where ij is the one such 00:18:06.700 --> 00:18:12.430 align:middle line:84% that the balls of radius R around yi is contained in Uij. 00:18:12.430 --> 00:18:15.990 align:middle line:90% 00:18:15.990 --> 00:18:19.830 align:middle line:84% And then we're done with the forward direction, right? 00:18:19.830 --> 00:18:21.570 align:middle line:84% Because we've gone from an open cover 00:18:21.570 --> 00:18:24.000 align:middle line:84% of our sequentially compact metric space 00:18:24.000 --> 00:18:25.470 align:middle line:90% and reached a finite subcover. 00:18:25.470 --> 00:18:28.200 align:middle line:90% 00:18:28.200 --> 00:18:30.810 align:middle line:84% So that shows that sequentially compact 00:18:30.810 --> 00:18:33.590 align:middle line:90% implies topologically compact. 00:18:33.590 --> 00:18:37.670 align:middle line:84% The other direction is going to be mildly harder, 00:18:37.670 --> 00:18:43.820 align:middle line:84% but it's still going to be essentially what we've 00:18:43.820 --> 00:18:45.290 align:middle line:90% been doing this entire time. 00:18:45.290 --> 00:18:47.990 align:middle line:84% We're going to assume that it's not sequentially compact, 00:18:47.990 --> 00:18:51.320 align:middle line:84% construct a sequence, and take the convergent subsequence, 00:18:51.320 --> 00:18:54.600 align:middle line:84% and reach a contradiction somehow. 00:18:54.600 --> 00:18:55.200 align:middle line:90% OK. 00:18:55.200 --> 00:18:59.155 align:middle line:84% So suppose topologically compact-- 00:18:59.155 --> 00:19:09.130 align:middle line:90% 00:19:09.130 --> 00:19:10.970 align:middle line:90% oh, I'm holding it backwards. 00:19:10.970 --> 00:19:11.960 align:middle line:90% Let me rewrite this. 00:19:11.960 --> 00:19:12.460 align:middle line:90% Compact. 00:19:12.460 --> 00:19:18.250 align:middle line:90% 00:19:18.250 --> 00:19:21.460 align:middle line:84% And let's suppose for the sake of contradiction-- 00:19:21.460 --> 00:19:28.640 align:middle line:84% assume for the sake of contradiction-- 00:19:28.640 --> 00:19:30.440 align:middle line:90% we're not sequentially compact. 00:19:30.440 --> 00:19:35.820 align:middle line:90% 00:19:35.820 --> 00:19:39.460 align:middle line:90% What we're going to do is-- 00:19:39.460 --> 00:19:41.080 align:middle line:84% or sorry, let me first state what 00:19:41.080 --> 00:19:43.300 align:middle line:84% does it mean to be not sequentially compact? 00:19:43.300 --> 00:19:49.060 align:middle line:84% Then there exists some sequence-- 00:19:49.060 --> 00:19:52.780 align:middle line:90% let's call it xn, of course-- 00:19:52.780 --> 00:19:55.480 align:middle line:90% with no convergent subsequence. 00:19:55.480 --> 00:20:00.600 align:middle line:90% 00:20:00.600 --> 00:20:04.620 align:middle line:84% Our goal is to go from this to an open cover of our space 00:20:04.620 --> 00:20:06.630 align:middle line:84% and show that there is no finite subcover 00:20:06.630 --> 00:20:08.940 align:middle line:90% to reach our contradiction. 00:20:08.940 --> 00:20:10.270 align:middle line:90% Well, how do we do that? 00:20:10.270 --> 00:20:12.240 align:middle line:84% Well first, we need to note some facts 00:20:12.240 --> 00:20:14.760 align:middle line:90% about convergent subsequences. 00:20:14.760 --> 00:20:23.640 align:middle line:90% Firstly, none of the xi's-- 00:20:23.640 --> 00:20:25.390 align:middle line:84% actually, I guess I shouldn't say firstly. 00:20:25.390 --> 00:20:28.330 align:middle line:84% Just note that none of the xi can 00:20:28.330 --> 00:20:29.590 align:middle line:90% appear infinitely many times. 00:20:29.590 --> 00:20:38.405 align:middle line:90% 00:20:38.405 --> 00:20:39.030 align:middle line:90% Actually, yeah. 00:20:39.030 --> 00:20:40.920 align:middle line:90% I'm going to call this firstly. 00:20:40.920 --> 00:20:42.460 align:middle line:90% Why is this true? 00:20:42.460 --> 00:20:45.625 align:middle line:84% Well, if any of the xi's appeared infinitely many times, 00:20:45.625 --> 00:20:47.250 align:middle line:84% then that would mean that we could just 00:20:47.250 --> 00:20:50.550 align:middle line:84% take a very trivial subsequence of xi's, and that 00:20:50.550 --> 00:20:53.112 align:middle line:90% would converge to xi. 00:20:53.112 --> 00:20:55.570 align:middle line:84% So we know that none of them appears infinitely many times. 00:20:55.570 --> 00:20:57.028 align:middle line:84% So we can just assume that each one 00:20:57.028 --> 00:20:59.100 align:middle line:90% appears finitely many times. 00:20:59.100 --> 00:21:02.730 align:middle line:84% But furthermore, we know that for all epsilon bigger-- 00:21:02.730 --> 00:21:05.100 align:middle line:84% nope, that's not what I meant to say. 00:21:05.100 --> 00:21:08.460 align:middle line:84% We know that there exists an epsilon n bigger 00:21:08.460 --> 00:21:14.910 align:middle line:84% than 0 such that the ball of radius epsilon n around xn 00:21:14.910 --> 00:21:20.770 align:middle line:84% is simply the set xn, where here I mean the single point xn. 00:21:20.770 --> 00:21:27.480 align:middle line:84% So I guess to be extra clear, I'll write this as xj and xj. 00:21:27.480 --> 00:21:28.900 align:middle line:90% Why is this true? 00:21:28.900 --> 00:21:30.960 align:middle line:84% Well, suppose for the sake of contradiction 00:21:30.960 --> 00:21:33.450 align:middle line:84% that for every epsilon bigger than 0, 00:21:33.450 --> 00:21:39.480 align:middle line:84% there exists some xi in a ball of radius epsilon around xj. 00:21:39.480 --> 00:21:41.640 align:middle line:84% Well then what that would tell us 00:21:41.640 --> 00:21:44.850 align:middle line:84% is that we could take a convergent subsequence, which 00:21:44.850 --> 00:21:47.520 align:middle line:90% will converge to xj. 00:21:47.520 --> 00:21:51.330 align:middle line:84% We just choose closer and closer points to xj. 00:21:51.330 --> 00:21:53.640 align:middle line:84% So we know that there must exist some epsilon n such 00:21:53.640 --> 00:21:55.720 align:middle line:90% that this is the case. 00:21:55.720 --> 00:22:00.430 align:middle line:84% So we're going to use this to create our open cover. 00:22:00.430 --> 00:22:04.560 align:middle line:84% We're going to let these be the open cover of uj. 00:22:04.560 --> 00:22:09.810 align:middle line:90% So let's call them ui, or uj's. 00:22:09.810 --> 00:22:10.620 align:middle line:90% Yeah. 00:22:10.620 --> 00:22:13.300 align:middle line:90% From j equals 1 to infinity. 00:22:13.300 --> 00:22:15.090 align:middle line:84% But this isn't quite an open cover. 00:22:15.090 --> 00:22:17.350 align:middle line:84% We don't know that we've fully covered x yet. 00:22:17.350 --> 00:22:24.240 align:middle line:84% So what we're going to do is let U0 00:22:24.240 --> 00:22:30.600 align:middle line:84% be equal to x not including any of the terms in our sequence. 00:22:30.600 --> 00:22:35.500 align:middle line:90% 00:22:35.500 --> 00:22:37.570 align:middle line:90% Now, why does this make sense? 00:22:37.570 --> 00:22:40.630 align:middle line:84% Or sorry, why is this an open set? 00:22:40.630 --> 00:22:44.450 align:middle line:84% Well, we know that the complement of this set 00:22:44.450 --> 00:22:46.400 align:middle line:84% contains all of its limit points, which 00:22:46.400 --> 00:22:49.370 align:middle line:84% means that it's going to be closed, as you're shown 00:22:49.370 --> 00:22:51.437 align:middle line:90% on your second problem set. 00:22:51.437 --> 00:22:53.270 align:middle line:84% So what this tells us is that the complement 00:22:53.270 --> 00:22:55.460 align:middle line:84% of the complement is therefore going to be open. 00:22:55.460 --> 00:22:59.180 align:middle line:90% So U0 is open. 00:22:59.180 --> 00:23:02.150 align:middle line:84% Therefore, we've reached an open cover of our metric space x. 00:23:02.150 --> 00:23:06.920 align:middle line:84% We have that x is contained-- oh, wrong direction-- 00:23:06.920 --> 00:23:12.650 align:middle line:84% contained in U0 union the infinite union of ui's or uj's. 00:23:12.650 --> 00:23:17.130 align:middle line:90% j equals 1 to infinity. 00:23:17.130 --> 00:23:21.320 align:middle line:84% Now, from here we want to reach a finite subcover, 00:23:21.320 --> 00:23:32.650 align:middle line:84% but notice that every finite subcover is 00:23:32.650 --> 00:23:42.650 align:middle line:84% going to be emitting infinitely many points, right? 00:23:42.650 --> 00:23:45.380 align:middle line:84% Because every finite subcover is going to have finitely many 00:23:45.380 --> 00:23:47.540 align:middle line:90% of these uj's. 00:23:47.540 --> 00:23:52.575 align:middle line:84% But what that tells you is that infinitely many of them-- 00:23:52.575 --> 00:23:54.950 align:middle line:84% sorry, let me double-check what I want to be saying here. 00:23:54.950 --> 00:23:58.700 align:middle line:90% 00:23:58.700 --> 00:23:59.330 align:middle line:90% Yeah. 00:23:59.330 --> 00:24:02.150 align:middle line:84% Because it's going to emit infinitely many points, 00:24:02.150 --> 00:24:05.120 align:middle line:90% this tells us that x-- 00:24:05.120 --> 00:24:16.473 align:middle line:84% or sorry, there is no finite subcover, 00:24:16.473 --> 00:24:18.140 align:middle line:84% which is our contradiction because we're 00:24:18.140 --> 00:24:20.570 align:middle line:90% assuming topologically compact. 00:24:20.570 --> 00:24:23.390 align:middle line:84% So we've shown that topologically compact implies 00:24:23.390 --> 00:24:24.620 align:middle line:90% sequentially compact. 00:24:24.620 --> 00:24:27.980 align:middle line:84% And that concludes the other direction of our proof. 00:24:27.980 --> 00:24:30.510 align:middle line:90% And thus, we're done. 00:24:30.510 --> 00:24:33.650 align:middle line:84% So this is essentially what we've 00:24:33.650 --> 00:24:36.020 align:middle line:84% set off to do from the very beginning, which is very 00:24:36.020 --> 00:24:39.790 align:middle line:90% motivated by metric spaces. 00:24:39.790 --> 00:24:41.740 align:middle line:84% But as you may guess from the fact 00:24:41.740 --> 00:24:47.753 align:middle line:84% that sequentially compact implies totally bounded, 00:24:47.753 --> 00:24:50.170 align:middle line:84% there's quite a bit more that we can actually be assuming. 00:24:50.170 --> 00:24:53.780 align:middle line:90% 00:24:53.780 --> 00:24:56.450 align:middle line:84% So what we're going to show for the rest of today 00:24:56.450 --> 00:24:59.750 align:middle line:84% is two more equivalent definitions of compactness 00:24:59.750 --> 00:25:01.280 align:middle line:90% on a metric space. 00:25:01.280 --> 00:25:04.670 align:middle line:84% And I'll again reiterate these are for a metric space, not 00:25:04.670 --> 00:25:10.040 align:middle line:84% inherently a topological space, as you might see in 18901. 00:25:10.040 --> 00:25:13.310 align:middle line:84% But this is the definition of compactness or equivalent 00:25:13.310 --> 00:25:15.950 align:middle line:84% definitions of compactness for a metric space. 00:25:15.950 --> 00:25:19.110 align:middle line:90% 00:25:19.110 --> 00:25:22.550 align:middle line:84% But before I do that, I want to first state some lemmas 00:25:22.550 --> 00:25:24.530 align:middle line:84% or some corollaries of these facts 00:25:24.530 --> 00:25:28.250 align:middle line:84% that we've shown so far because they're deeply important. 00:25:28.250 --> 00:25:30.362 align:middle line:84% Compactness has some very deep results. 00:25:30.362 --> 00:25:31.820 align:middle line:84% And I know that this is going to be 00:25:31.820 --> 00:25:33.450 align:middle line:90% a very proof-intensive class. 00:25:33.450 --> 00:25:36.237 align:middle line:84% So I want to first start by stating some facts 00:25:36.237 --> 00:25:37.820 align:middle line:84% that are a little bit easier to prove, 00:25:37.820 --> 00:25:40.460 align:middle line:90% but also mildly intuitive. 00:25:40.460 --> 00:25:47.610 align:middle line:84% So first, recall a function f from x to y, 00:25:47.610 --> 00:25:58.190 align:middle line:84% where x and y are metric spaces, is continuous if and only 00:25:58.190 --> 00:26:09.185 align:middle line:84% if, for every u open in y, f inverse of u is open in x. 00:26:09.185 --> 00:26:12.020 align:middle line:90% 00:26:12.020 --> 00:26:13.770 align:middle line:84% Sorry, I know I didn't fully write it out, 00:26:13.770 --> 00:26:16.140 align:middle line:84% but I at least said it verbally. f inverse of you 00:26:16.140 --> 00:26:17.580 align:middle line:90% is going to be open in x. 00:26:17.580 --> 00:26:19.800 align:middle line:84% This is the definition of continuous. 00:26:19.800 --> 00:26:26.110 align:middle line:84% And what we're going to do is therefore show lemma. 00:26:26.110 --> 00:26:33.850 align:middle line:84% If k is a compact subset of x, where here the double subsets 00:26:33.850 --> 00:26:36.930 align:middle line:90% implies compact-- 00:26:36.930 --> 00:26:39.030 align:middle line:90% so recall that notation-- 00:26:39.030 --> 00:26:46.830 align:middle line:84% we're going to note that f of k is compact in y. 00:26:46.830 --> 00:26:51.570 align:middle line:84% Over here f is a continuous map from x to y. 00:26:51.570 --> 00:26:55.572 align:middle line:90% 00:26:55.572 --> 00:26:57.280 align:middle line:84% So this is our first statement that we're 00:26:57.280 --> 00:26:59.753 align:middle line:90% going to prove today. 00:26:59.753 --> 00:27:01.420 align:middle line:84% Or sorry, the first statement that we're 00:27:01.420 --> 00:27:04.990 align:middle line:90% going to prove right now. 00:27:04.990 --> 00:27:11.220 align:middle line:84% Proof-- well, we want to somehow reach an open cover of k. 00:27:11.220 --> 00:27:30.340 align:middle line:84% So let Ui be an open cover of f of k in y. 00:27:30.340 --> 00:27:32.110 align:middle line:90% Then what do we know? 00:27:32.110 --> 00:27:39.860 align:middle line:84% Well, what this will imply is that f inverse of Ui 00:27:39.860 --> 00:27:41.390 align:middle line:90% will be a finite-- 00:27:41.390 --> 00:27:51.770 align:middle line:84% sorry, not finite-- will be an open cover of k. 00:27:51.770 --> 00:27:53.450 align:middle line:84% The fact that the inverse images cover 00:27:53.450 --> 00:27:56.120 align:middle line:84% k is by the very definition of f of k, 00:27:56.120 --> 00:27:58.070 align:middle line:84% but how do we know that they're open? 00:27:58.070 --> 00:28:00.300 align:middle line:84% Well, this is from the fact that it's continuous. 00:28:00.300 --> 00:28:04.070 align:middle line:84% These are the inverse images of open sets. 00:28:04.070 --> 00:28:09.110 align:middle line:84% And so what we know is that we can reach a finite subcover-- 00:28:09.110 --> 00:28:20.460 align:middle line:84% a finite subcover given by f inverse Ui from, 00:28:20.460 --> 00:28:24.030 align:middle line:90% let's say, i equals 1 to k. 00:28:24.030 --> 00:28:27.000 align:middle line:84% And what we're going to do is map this back to f of k. 00:28:27.000 --> 00:28:37.520 align:middle line:84% Notice that then f of f inverse of Ui 00:28:37.520 --> 00:28:43.760 align:middle line:84% equals Ui, which is therefore going 00:28:43.760 --> 00:28:46.040 align:middle line:90% to give us our open-- sorry. 00:28:46.040 --> 00:28:48.600 align:middle line:90% How should I say this? 00:28:48.600 --> 00:28:49.590 align:middle line:90% How should I say this? 00:28:49.590 --> 00:28:53.010 align:middle line:84% This is not necessarily directly true. 00:28:53.010 --> 00:29:00.450 align:middle line:84% What I mean to say is that then Ui from i equals 1 to k 00:29:00.450 --> 00:29:09.050 align:middle line:90% is open cover of f of k. 00:29:09.050 --> 00:29:10.560 align:middle line:90% How do we know this? 00:29:10.560 --> 00:29:13.010 align:middle line:84% Well, we know that the Ui's are such that-- 00:29:13.010 --> 00:29:17.060 align:middle line:84% or sorry, the f inverse of Ui's cover k. 00:29:17.060 --> 00:29:21.200 align:middle line:84% So therefore, the images of them must cover f of k. 00:29:21.200 --> 00:29:23.990 align:middle line:84% And so we've gone from an open cover of f of k 00:29:23.990 --> 00:29:28.400 align:middle line:84% to a finite subcover, which shows that it's compact in y. 00:29:28.400 --> 00:29:29.660 align:middle line:90% So that is our proof. 00:29:29.660 --> 00:29:32.150 align:middle line:84% That shows that the image of compact sets 00:29:32.150 --> 00:29:36.560 align:middle line:84% is going to be compact under continuity. 00:29:36.560 --> 00:29:39.230 align:middle line:84% And using this and the fact that we have sequential compactness 00:29:39.230 --> 00:29:43.010 align:middle line:84% is going to allow us to state some very helpful lemmas. 00:29:43.010 --> 00:29:47.540 align:middle line:90% So lemma-- or sorry, not lemma-- 00:29:47.540 --> 00:29:48.220 align:middle line:90% corollary. 00:29:48.220 --> 00:29:53.430 align:middle line:90% 00:29:53.430 --> 00:30:03.330 align:middle line:84% On that f from x to, let's say, the real numbers, continuous 00:30:03.330 --> 00:30:21.360 align:middle line:84% has a max and a min achieved on compact sets, by which I 00:30:21.360 --> 00:30:25.590 align:middle line:84% mean for every compact subset of x, the image of it to R 00:30:25.590 --> 00:30:28.200 align:middle line:84% is going to have a maximum and a minimum achieved. 00:30:28.200 --> 00:30:31.540 align:middle line:84% And the proof is very short at this point. 00:30:31.540 --> 00:30:35.310 align:middle line:84% We know that f of k is going to be 00:30:35.310 --> 00:30:37.858 align:middle line:84% compact in R, which implies that it's 00:30:37.858 --> 00:30:39.150 align:middle line:90% going to be closed and bounded. 00:30:39.150 --> 00:30:44.610 align:middle line:90% 00:30:44.610 --> 00:30:46.740 align:middle line:84% Closed and bounded, which you can 00:30:46.740 --> 00:30:48.960 align:middle line:84% think of as sequentially compact in the same way, 00:30:48.960 --> 00:30:52.230 align:middle line:84% implies that we can find a maximum and a minimum of f 00:30:52.230 --> 00:30:53.550 align:middle line:90% of k. 00:30:53.550 --> 00:30:56.020 align:middle line:84% This is essentially just the extreme value theorem. 00:30:56.020 --> 00:30:58.770 align:middle line:84% But it's much more general than the extreme value theorem. 00:30:58.770 --> 00:31:00.420 align:middle line:84% Before, the extreme value theorem 00:31:00.420 --> 00:31:03.450 align:middle line:84% was a map from the real numbers, or at least a compact subset, 00:31:03.450 --> 00:31:06.070 align:middle line:84% like a to b, to the real numbers. 00:31:06.070 --> 00:31:09.120 align:middle line:84% Now, all of a sudden, we have the extreme value theorem 00:31:09.120 --> 00:31:13.260 align:middle line:90% for metric spaces. 00:31:13.260 --> 00:31:17.080 align:middle line:90% This fact is extremely helpful. 00:31:17.080 --> 00:31:20.430 align:middle line:84% I cannot iterate how important it's going to be in the long 00:31:20.430 --> 00:31:22.290 align:middle line:84% term if you keep taking analysis courses, 00:31:22.290 --> 00:31:25.680 align:middle line:84% but the fact that images of compact sets are going to be 00:31:25.680 --> 00:31:30.510 align:middle line:84% compact is very, very helpful, and helps us with finding 00:31:30.510 --> 00:31:31.860 align:middle line:90% a maximum and a minimum. 00:31:31.860 --> 00:31:34.590 align:middle line:84% In fact, one other way to think of this theorem 00:31:34.590 --> 00:31:37.140 align:middle line:84% is just to take a sequence that converges 00:31:37.140 --> 00:31:39.540 align:middle line:84% to the maximum and converges to the minimum. 00:31:39.540 --> 00:31:42.098 align:middle line:84% And there has to be a convergent subsequence of that sequence 00:31:42.098 --> 00:31:43.515 align:middle line:84% because it's sequentially compact. 00:31:43.515 --> 00:31:46.120 align:middle line:90% 00:31:46.120 --> 00:31:50.610 align:middle line:84% Besides this, I also want to note-- 00:31:50.610 --> 00:32:07.230 align:middle line:84% corollary-- maps f given x is compact, 00:32:07.230 --> 00:32:16.800 align:middle line:84% f from x to R continuous is going to be bounded. 00:32:16.800 --> 00:32:20.800 align:middle line:90% 00:32:20.800 --> 00:32:23.530 align:middle line:84% This is the second corollary, right? 00:32:23.530 --> 00:32:25.300 align:middle line:84% Before, we were using closedness. 00:32:25.300 --> 00:32:27.440 align:middle line:84% Right now, we're using boundedness. 00:32:27.440 --> 00:32:30.820 align:middle line:84% So this is very helpful just in order 00:32:30.820 --> 00:32:34.190 align:middle line:84% to be able to conceptualize what's happening. 00:32:34.190 --> 00:32:38.560 align:middle line:84% So the proof is already complete by before. 00:32:38.560 --> 00:32:42.040 align:middle line:84% So I will stop there for the moment being. 00:32:42.040 --> 00:32:43.540 align:middle line:84% Now, we have one more theorem that I 00:32:43.540 --> 00:32:49.240 align:middle line:84% want to show that will have some implications that 00:32:49.240 --> 00:32:50.840 align:middle line:90% will be conceptually helpful. 00:32:50.840 --> 00:32:53.840 align:middle line:84% But before I do that, I'm going to drink some water. 00:32:53.840 --> 00:32:57.520 align:middle line:84% The next theorem is known as Cantor's intersection theorem. 00:32:57.520 --> 00:33:01.420 align:middle line:84% It's one that tells us things about nested sequences 00:33:01.420 --> 00:33:03.850 align:middle line:90% of compact sets. 00:33:03.850 --> 00:33:06.020 align:middle line:90% And it's helpful conceptually. 00:33:06.020 --> 00:33:07.790 align:middle line:90% It's used all the time. 00:33:07.790 --> 00:33:10.192 align:middle line:84% In fact, if you were to take a complex analysis class, 00:33:10.192 --> 00:33:11.650 align:middle line:84% it's one of the first things that's 00:33:11.650 --> 00:33:16.900 align:middle line:84% used to show convergence of line integrals 00:33:16.900 --> 00:33:20.460 align:middle line:90% for holomorphic functions. 00:33:20.460 --> 00:33:21.830 align:middle line:90% So what's the statement? 00:33:21.830 --> 00:33:25.580 align:middle line:90% 00:33:25.580 --> 00:33:29.275 align:middle line:84% Theorem-- this is known as Cantor's intersection theorem. 00:33:29.275 --> 00:33:40.660 align:middle line:90% 00:33:40.660 --> 00:33:49.830 align:middle line:84% So given K1 containing K2 containing 00:33:49.830 --> 00:33:58.290 align:middle line:84% K3, so on and so forth, each compact, each nonempty 00:33:58.290 --> 00:33:58.980 align:middle line:90% and compact-- 00:33:58.980 --> 00:34:03.400 align:middle line:90% 00:34:03.400 --> 00:34:07.810 align:middle line:84% by compact, I mean topologically or sequentially, which we know. 00:34:07.810 --> 00:34:11.300 align:middle line:90% That's a simple fact that-- 00:34:11.300 --> 00:34:13.130 align:middle line:90% let me reiterate. 00:34:13.130 --> 00:34:17.000 align:middle line:84% Buy nonempty and compact, I mean sequentially or topologically 00:34:17.000 --> 00:34:19.400 align:middle line:84% because they're the equivalent on a metric space. 00:34:19.400 --> 00:34:21.860 align:middle line:84% And here I'm assuming that the compact subsets 00:34:21.860 --> 00:34:23.180 align:middle line:90% are in a metric space. 00:34:23.180 --> 00:34:25.889 align:middle line:90% 00:34:25.889 --> 00:34:32.480 align:middle line:84% Then the intersection of the ki's from i 00:34:32.480 --> 00:34:37.000 align:middle line:84% equals 1 to infinity is nonempty. 00:34:37.000 --> 00:34:39.830 align:middle line:90% How do we prove this? 00:34:39.830 --> 00:34:44.179 align:middle line:84% Well, we prove this by taking a sequence of terms in ki's. 00:34:44.179 --> 00:34:52.300 align:middle line:84% So proof-- let wi be in ki for all i. 00:34:52.300 --> 00:34:54.197 align:middle line:84% And we know that there exists such an xi 00:34:54.197 --> 00:34:55.239 align:middle line:90% because they're nonempty. 00:34:55.239 --> 00:34:58.930 align:middle line:84% So right off the bat, we're using that hypothesis. 00:34:58.930 --> 00:35:03.380 align:middle line:84% And what we want to note is then that there exists-- 00:35:03.380 --> 00:35:06.260 align:middle line:84% or sorry, let's focus on K1 for now. 00:35:06.260 --> 00:35:12.875 align:middle line:90% Notice xi is in K1 for all i. 00:35:12.875 --> 00:35:15.750 align:middle line:90% 00:35:15.750 --> 00:35:18.060 align:middle line:84% This follows from the fact that it's a nested sequence 00:35:18.060 --> 00:35:20.430 align:middle line:90% of compact sets. 00:35:20.430 --> 00:35:31.780 align:middle line:84% So therefore, there exists a convergent subsequence xnk 00:35:31.780 --> 00:35:37.300 align:middle line:90% converging to sum a in K1. 00:35:37.300 --> 00:35:39.950 align:middle line:84% And what you can continue to show 00:35:39.950 --> 00:35:42.110 align:middle line:84% is that it's going to continue to convergence 00:35:42.110 --> 00:35:45.300 align:middle line:90% into a in every ki. 00:35:45.300 --> 00:36:04.030 align:middle line:84% So secondly, notice-- xi from i equal to 2 to infinity 00:36:04.030 --> 00:36:09.580 align:middle line:84% as opposed to including 1 is going to be a subset of K2. 00:36:09.580 --> 00:36:12.220 align:middle line:84% And therefore, there exists a convergent subsequence. 00:36:12.220 --> 00:36:15.470 align:middle line:84% And the convergent subsequence has a unique limit point, 00:36:15.470 --> 00:36:15.970 align:middle line:90% right? 00:36:15.970 --> 00:36:17.740 align:middle line:84% This is what we said on the second day. 00:36:17.740 --> 00:36:21.190 align:middle line:84% Convergent sequences have equivalent or have the same-- 00:36:21.190 --> 00:36:26.282 align:middle line:90% 00:36:26.282 --> 00:36:27.365 align:middle line:90% how do I want to say this? 00:36:27.365 --> 00:36:30.160 align:middle line:90% 00:36:30.160 --> 00:36:31.730 align:middle line:90% Have the same limit points. 00:36:31.730 --> 00:36:35.230 align:middle line:90% So therefore, a will be in K2. 00:36:35.230 --> 00:36:37.960 align:middle line:84% And we can continue this process over and over again, 00:36:37.960 --> 00:36:42.620 align:middle line:84% ending up with a single point at least in all of the ki's. 00:36:42.620 --> 00:36:45.940 align:middle line:90% So this concludes our proof. 00:36:45.940 --> 00:36:48.700 align:middle line:84% And this is used in complex analysis, 00:36:48.700 --> 00:36:53.650 align:middle line:84% as I stated, to go from the sequence of triangles, 00:36:53.650 --> 00:36:54.150 align:middle line:90% let's say. 00:36:54.150 --> 00:36:56.372 align:middle line:90% 00:36:56.372 --> 00:36:58.330 align:middle line:84% So let's say we have the sequence of triangles, 00:36:58.330 --> 00:37:01.870 align:middle line:84% and we know that there exists a point in every one of them. 00:37:01.870 --> 00:37:04.120 align:middle line:84% We can keep reiterating this pattern 00:37:04.120 --> 00:37:06.550 align:middle line:84% and end up with a point in the limit. 00:37:06.550 --> 00:37:09.420 align:middle line:90% 00:37:09.420 --> 00:37:12.000 align:middle line:84% So we know that the limits of the intersection 00:37:12.000 --> 00:37:13.815 align:middle line:90% is going to be nonempty. 00:37:13.815 --> 00:37:17.230 align:middle line:84% So that's how it's used in complex analysis. 00:37:17.230 --> 00:37:20.840 align:middle line:84% Of course, we can't discuss this more fully 00:37:20.840 --> 00:37:23.630 align:middle line:84% now because it requires a discussion that 00:37:23.630 --> 00:37:26.980 align:middle line:84% takes numerous days of complex analysis to prove. 00:37:26.980 --> 00:37:30.000 align:middle line:84% But if you take complex analysis, 00:37:30.000 --> 00:37:33.090 align:middle line:84% this will be one of the first things you see. 00:37:33.090 --> 00:37:34.728 align:middle line:90% OK. 00:37:34.728 --> 00:37:36.770 align:middle line:84% So what are we going to do with this new theorem? 00:37:36.770 --> 00:37:39.830 align:middle line:84% Well, we're going to define a new property. 00:37:39.830 --> 00:37:45.685 align:middle line:84% Definition-- a collection of closed sets-- 00:37:45.685 --> 00:37:51.060 align:middle line:90% 00:37:51.060 --> 00:37:53.385 align:middle line:84% notice here that I'm assuming closed and not compact-- 00:37:53.385 --> 00:37:56.600 align:middle line:90% 00:37:56.600 --> 00:38:20.530 align:middle line:84% is or has the finite intersection property 00:38:20.530 --> 00:38:25.980 align:middle line:90% if every finite subsection-- 00:38:25.980 --> 00:38:30.930 align:middle line:90% 00:38:30.930 --> 00:38:35.590 align:middle line:84% i.e. if I take finitely many of the closed sets-- 00:38:35.590 --> 00:38:39.520 align:middle line:84% we're going to assume that it has nonempty n intersection. 00:38:39.520 --> 00:38:47.280 align:middle line:90% 00:38:47.280 --> 00:38:48.750 align:middle line:84% We know that this is already going 00:38:48.750 --> 00:38:50.460 align:middle line:84% to be true for compact sets because 00:38:50.460 --> 00:38:52.552 align:middle line:84% the infinite intersection of them is nonempty. 00:38:52.552 --> 00:38:54.510 align:middle line:84% But this statement is a little bit more general 00:38:54.510 --> 00:38:57.282 align:middle line:84% in that we're only assuming that it's going to be closed. 00:38:57.282 --> 00:38:59.490 align:middle line:84% And so we're assuming that every finite subcollection 00:38:59.490 --> 00:39:03.647 align:middle line:90% has a nonempty intersection. 00:39:03.647 --> 00:39:05.980 align:middle line:84% And so what we're going to show is that in fact, there's 00:39:05.980 --> 00:39:08.350 align:middle line:84% a relationship between the finite intersection 00:39:08.350 --> 00:39:12.400 align:middle line:84% property and topologically and sequentially compact. 00:39:12.400 --> 00:39:14.470 align:middle line:84% That's going to be our huge theorem for today. 00:39:14.470 --> 00:39:18.310 align:middle line:84% It's going to take a little bit to prove, 00:39:18.310 --> 00:39:21.670 align:middle line:84% but will be deeply powerful once we're done. 00:39:21.670 --> 00:39:24.770 align:middle line:90% 00:39:24.770 --> 00:39:28.310 align:middle line:84% Our theorem is that the following are equivalent. 00:39:28.310 --> 00:39:31.600 align:middle line:90% 00:39:31.600 --> 00:39:36.190 align:middle line:84% The following are equivalent, which I'll write out more fully 00:39:36.190 --> 00:39:49.050 align:middle line:84% in case you haven't seen this abbreviation before 00:39:49.050 --> 00:39:50.630 align:middle line:90% on a metric space x. 00:39:50.630 --> 00:39:59.560 align:middle line:90% 00:39:59.560 --> 00:40:05.045 align:middle line:90% One, x is topologically compact. 00:40:05.045 --> 00:40:08.180 align:middle line:90% 00:40:08.180 --> 00:40:09.110 align:middle line:90% Topologically. 00:40:09.110 --> 00:40:18.620 align:middle line:90% 00:40:18.620 --> 00:40:25.630 align:middle line:84% Two, x is sequentially compact, which makes sense. 00:40:25.630 --> 00:40:27.840 align:middle line:84% We've already shown this implication, 00:40:27.840 --> 00:40:31.960 align:middle line:84% but we're going to show two more properties right now. 00:40:31.960 --> 00:40:35.200 align:middle line:90% Three, x is totally bounded. 00:40:35.200 --> 00:40:39.450 align:middle line:90% 00:40:39.450 --> 00:40:41.450 align:middle line:84% We've already shown sequentially compact implies 00:40:41.450 --> 00:40:44.690 align:middle line:84% totally bounded, but there's one more thing we need to include. 00:40:44.690 --> 00:40:46.565 align:middle line:84% We need to include that it's Cauchy complete. 00:40:46.565 --> 00:40:53.460 align:middle line:90% 00:40:53.460 --> 00:40:56.840 align:middle line:84% And finally, our fourth property is 00:40:56.840 --> 00:40:59.147 align:middle line:84% going to be the following, which I'll state out clearly 00:40:59.147 --> 00:41:00.980 align:middle line:84% because it's one of the more confusing ones. 00:41:00.980 --> 00:41:04.500 align:middle line:90% 00:41:04.500 --> 00:41:17.580 align:middle line:84% Every collection of closed sets of closed subsets 00:41:17.580 --> 00:41:36.730 align:middle line:84% of x with the finite intersection property 00:41:36.730 --> 00:41:38.680 align:middle line:90% has nonempty intersection. 00:41:38.680 --> 00:41:49.650 align:middle line:90% 00:41:49.650 --> 00:41:51.810 align:middle line:84% Here the difference is that we know 00:41:51.810 --> 00:41:55.020 align:middle line:84% that every finite subcollection has nonempty intersection, 00:41:55.020 --> 00:41:56.940 align:middle line:84% but what this theorem is telling you 00:41:56.940 --> 00:42:00.180 align:middle line:84% is that the intersection of all of the closed subsets, 00:42:00.180 --> 00:42:02.310 align:middle line:84% if they have the finite intersection property, 00:42:02.310 --> 00:42:04.050 align:middle line:90% is also going to be nonempty. 00:42:04.050 --> 00:42:06.720 align:middle line:90% 00:42:06.720 --> 00:42:10.170 align:middle line:84% Now, what we're going to do is show that 1 is true if only 00:42:10.170 --> 00:42:11.145 align:middle line:90% if 4 is true. 00:42:11.145 --> 00:42:14.375 align:middle line:90% 00:42:14.375 --> 00:42:15.750 align:middle line:84% Let me write this out more fully. 00:42:15.750 --> 00:42:19.650 align:middle line:84% We've already shown that 1 is true if an only if 2 is true. 00:42:19.650 --> 00:42:22.770 align:middle line:84% What we're now going to do is show that 1 is true if 00:42:22.770 --> 00:42:28.650 align:middle line:84% and only if 4 is true, and then show that 2 is true if and only 00:42:28.650 --> 00:42:30.870 align:middle line:84% if 3 is true because we've already shown sequentially 00:42:30.870 --> 00:42:33.100 align:middle line:90% compact implies totally bounded. 00:42:33.100 --> 00:42:36.405 align:middle line:84% So we would just have to show Cauchy completeness. 00:42:36.405 --> 00:42:38.280 align:middle line:84% And then that's going to be the end of today. 00:42:38.280 --> 00:42:41.160 align:middle line:84% We're going to show these four properties of compact sets, 00:42:41.160 --> 00:42:45.940 align:middle line:84% and then we're going to end today's discussion. 00:42:45.940 --> 00:42:49.635 align:middle line:84% So let's start with showing that 1 is true implies 4 is true. 00:42:49.635 --> 00:43:01.820 align:middle line:90% 00:43:01.820 --> 00:43:05.080 align:middle line:84% So the proof is going to start with supposing, 00:43:05.080 --> 00:43:07.600 align:middle line:84% for the sake of contradiction, there exists 00:43:07.600 --> 00:43:10.060 align:middle line:84% a collection of closed sets with the finite intersection 00:43:10.060 --> 00:43:13.060 align:middle line:84% property, but with empty intersection. 00:43:13.060 --> 00:43:21.730 align:middle line:90% So suppose there exists ci-- 00:43:21.730 --> 00:43:24.040 align:middle line:90% i equals 1 to infinity-- 00:43:24.040 --> 00:43:35.930 align:middle line:84% closed with the finite intersection property, 00:43:35.930 --> 00:43:38.390 align:middle line:84% i.e. intersection of finite subcollections 00:43:38.390 --> 00:43:48.320 align:middle line:84% is going to be nonempty, and the intersection of all the ci's is 00:43:48.320 --> 00:43:49.250 align:middle line:90% empty. 00:43:49.250 --> 00:43:52.645 align:middle line:84% That's going to be our assumption for contradiction. 00:43:52.645 --> 00:43:54.770 align:middle line:84% Well, knowing that the intersection of these closed 00:43:54.770 --> 00:43:56.540 align:middle line:84% sets is going to be empty, what do we 00:43:56.540 --> 00:43:58.070 align:middle line:90% know about the complements? 00:43:58.070 --> 00:44:01.340 align:middle line:84% Well, then we know that x, which is the complement of the empty 00:44:01.340 --> 00:44:04.850 align:middle line:84% set, is going to be the complement of the intersection 00:44:04.850 --> 00:44:07.130 align:middle line:90% of ci's. 00:44:07.130 --> 00:44:10.010 align:middle line:84% But then we know by De Morgan's laws, which we introduced 00:44:10.010 --> 00:44:13.290 align:middle line:84% on the second day in general theory, 00:44:13.290 --> 00:44:17.540 align:middle line:84% it's going to be the union of the complements of ci's from i 00:44:17.540 --> 00:44:20.060 align:middle line:90% equals 1 to infinity. 00:44:20.060 --> 00:44:21.680 align:middle line:90% What do we know then? 00:44:21.680 --> 00:44:24.740 align:middle line:84% Well, the complement of a closed set is going to be open. 00:44:24.740 --> 00:44:27.710 align:middle line:84% So what we've gone from is this collection of closed 00:44:27.710 --> 00:44:31.040 align:middle line:90% sets to an open cover of x. 00:44:31.040 --> 00:44:41.770 align:middle line:84% Therefore, there exists a finite subcover given by, 00:44:41.770 --> 00:44:46.210 align:middle line:84% let's say, the union from i equals 1 to k of ci complement. 00:44:46.210 --> 00:44:49.360 align:middle line:90% 00:44:49.360 --> 00:44:51.430 align:middle line:84% And what we're going to show is then 00:44:51.430 --> 00:44:54.250 align:middle line:84% that it fails to have the finite intersection property. 00:44:54.250 --> 00:45:00.920 align:middle line:84% So notice then that the intersection of ci from i 00:45:00.920 --> 00:45:05.510 align:middle line:84% equals 1 to k is going to be the same 00:45:05.510 --> 00:45:09.800 align:middle line:84% as the union of the complements from i 00:45:09.800 --> 00:45:13.790 align:middle line:84% equals 1 to k, which we know is x. 00:45:13.790 --> 00:45:18.687 align:middle line:84% So what this tells us is that the complement of this-- 00:45:18.687 --> 00:45:20.270 align:middle line:84% sorry, that's not what I meant to say. 00:45:20.270 --> 00:45:27.400 align:middle line:90% 00:45:27.400 --> 00:45:29.020 align:middle line:90% Let me reiterate. 00:45:29.020 --> 00:45:37.650 align:middle line:84% What we know then is that x is equal to the union of ci 00:45:37.650 --> 00:45:42.560 align:middle line:84% complement from i equals 1 to k, which 00:45:42.560 --> 00:45:49.340 align:middle line:84% is, of course, equal to the intersection from n equals 1-- 00:45:49.340 --> 00:45:56.222 align:middle line:84% oh, OK, I see-- from i equals 1 to k ci complement. 00:45:56.222 --> 00:45:58.180 align:middle line:84% So we've gone from the union of the complements 00:45:58.180 --> 00:46:01.090 align:middle line:84% to the complement of the intersection. 00:46:01.090 --> 00:46:04.660 align:middle line:84% And then this implies that the intersection from i 00:46:04.660 --> 00:46:09.010 align:middle line:84% equals 1 to k of ci must be the empty set, which 00:46:09.010 --> 00:46:11.560 align:middle line:84% is a contradiction because we're assuming that it has 00:46:11.560 --> 00:46:14.320 align:middle line:84% the finite intersection property, which again means 00:46:14.320 --> 00:46:18.880 align:middle line:84% that every finite subcollection has a nonempty intersection. 00:46:18.880 --> 00:46:20.350 align:middle line:90% So this is our contradiction. 00:46:20.350 --> 00:46:23.500 align:middle line:84% And we've shown that topologically compact implies 00:46:23.500 --> 00:46:25.540 align:middle line:84% that every collection of closed sets 00:46:25.540 --> 00:46:27.880 align:middle line:84% has with the finite intersection property 00:46:27.880 --> 00:46:31.880 align:middle line:90% has nonempty intersection. 00:46:31.880 --> 00:46:35.190 align:middle line:84% The proof of the other direction is going to be very similar. 00:46:35.190 --> 00:46:37.670 align:middle line:84% We're going to show that it's topologically compact given 00:46:37.670 --> 00:46:39.410 align:middle line:90% property 4. 00:46:39.410 --> 00:46:42.530 align:middle line:84% So we're going to show that 4 implies 1. 00:46:42.530 --> 00:46:45.450 align:middle line:90% 00:46:45.450 --> 00:46:57.540 align:middle line:84% Suppose z Ui's are an open cover of x. 00:46:57.540 --> 00:47:04.760 align:middle line:90% 00:47:04.760 --> 00:47:07.570 align:middle line:84% What we want to go from, then, is 00:47:07.570 --> 00:47:10.700 align:middle line:84% we want to use this to find information about closed sets 00:47:10.700 --> 00:47:12.490 align:middle line:90% so that we can apply property 4. 00:47:12.490 --> 00:47:18.700 align:middle line:84% Well, what we'll do is let Ui complement these ci, which 00:47:18.700 --> 00:47:20.470 align:middle line:90% are going to be closed. 00:47:20.470 --> 00:47:21.970 align:middle line:84% How do you know that they're closed? 00:47:21.970 --> 00:47:25.000 align:middle line:84% Because the complement is going to be the Ui's, which are open, 00:47:25.000 --> 00:47:27.550 align:middle line:84% so that implies that it's closed. 00:47:27.550 --> 00:47:31.930 align:middle line:84% And so what we're going to show is that the ci's 00:47:31.930 --> 00:47:34.670 align:middle line:90% have finite intersection. 00:47:34.670 --> 00:47:37.580 align:middle line:84% So of the final intersection property-- 00:47:37.580 --> 00:47:45.510 align:middle line:90% 00:47:45.510 --> 00:47:47.010 align:middle line:84% assume for the sake of contradiction 00:47:47.010 --> 00:47:50.950 align:middle line:84% that they don't have the finite intersection property-- 00:47:50.950 --> 00:47:53.800 align:middle line:90% for the sake of contradiction-- 00:47:53.800 --> 00:48:00.730 align:middle line:84% that the intersection of ci1 cik is empty, i.e. 00:48:00.730 --> 00:48:04.340 align:middle line:84% it doesn't have the finite intersection property. 00:48:04.340 --> 00:48:15.740 align:middle line:84% Well then we'll have is that the union from i equals 1 to k 00:48:15.740 --> 00:48:22.010 align:middle line:84% of the Ui's is equal to the intersection of the ci k, 00:48:22.010 --> 00:48:30.210 align:middle line:84% ci n from n equals 1 to k complement complement, 00:48:30.210 --> 00:48:34.280 align:middle line:84% which is, of course, then the complement of the intersection 00:48:34.280 --> 00:48:36.335 align:middle line:90% of the cosets, which is x. 00:48:36.335 --> 00:48:43.070 align:middle line:90% 00:48:43.070 --> 00:48:44.810 align:middle line:84% Do I want to read you this proof? 00:48:44.810 --> 00:48:46.100 align:middle line:90% I will read you this proof. 00:48:46.100 --> 00:48:48.440 align:middle line:84% I want to rewrite this out more carefully because we 00:48:48.440 --> 00:48:53.630 align:middle line:90% have two nested contradictions. 00:48:53.630 --> 00:49:04.670 align:middle line:84% So firstly, assume for the sake of contradiction that there is 00:49:04.670 --> 00:49:06.300 align:middle line:90% no finite subcover of the Ui's. 00:49:06.300 --> 00:49:10.530 align:middle line:90% 00:49:10.530 --> 00:49:17.775 align:middle line:90% Finite subcover of the Ui's-- 00:49:17.775 --> 00:49:20.475 align:middle line:90% 00:49:20.475 --> 00:49:21.037 align:middle line:90% i.e. 00:49:21.037 --> 00:49:22.870 align:middle line:84% we're assuming for the sake of contradiction 00:49:22.870 --> 00:49:25.210 align:middle line:84% that x is not topologically compact. 00:49:25.210 --> 00:49:28.360 align:middle line:84% And furthermore, we're going to assume 00:49:28.360 --> 00:49:37.710 align:middle line:84% for the sake of contradiction that the ci's 00:49:37.710 --> 00:49:41.190 align:middle line:84% don't have the finite intersection property. 00:49:41.190 --> 00:49:46.860 align:middle line:84% Therefore, because we want to show 00:49:46.860 --> 00:49:49.050 align:middle line:84% that they do have the finite intersection property, 00:49:49.050 --> 00:49:56.400 align:middle line:84% so the intersection from n equals 1 to k of ci n is empty. 00:49:56.400 --> 00:49:58.170 align:middle line:84% That's what we're going to assume. 00:49:58.170 --> 00:49:59.680 align:middle line:90% Well, then what do we know? 00:49:59.680 --> 00:50:02.940 align:middle line:84% We know then that the union of the Ui's-- 00:50:02.940 --> 00:50:09.150 align:middle line:90% Ui n from n equals 1 to k-- 00:50:09.150 --> 00:50:13.020 align:middle line:84% is the intersection of the complements-- 00:50:13.020 --> 00:50:17.790 align:middle line:90% complement from n equals 1 to k. 00:50:17.790 --> 00:50:19.740 align:middle line:84% But then this is going to be all of x, right? 00:50:19.740 --> 00:50:22.650 align:middle line:84% Because it's a complement of the empty set. 00:50:22.650 --> 00:50:24.900 align:middle line:84% And that's a contradiction because we're assuming that 00:50:24.900 --> 00:50:28.200 align:middle line:84% there's no finite subcover of the Ui's. 00:50:28.200 --> 00:50:37.820 align:middle line:84% So what we know then is that the ci's 00:50:37.820 --> 00:50:51.650 align:middle line:84% have the finite intersection property, i.e. 00:50:51.650 --> 00:50:54.260 align:middle line:84% the intersection of all of the ci's from i 00:50:54.260 --> 00:50:58.820 align:middle line:84% equals 1 to infinity is nonempty. 00:50:58.820 --> 00:50:59.972 align:middle line:90% So what do we know? 00:50:59.972 --> 00:51:02.470 align:middle line:84% We know then that the complement of these 00:51:02.470 --> 00:51:04.660 align:middle line:90% is going to be not all of x. 00:51:04.660 --> 00:51:09.490 align:middle line:84% So we know that x, which is the complement of the empty set, 00:51:09.490 --> 00:51:13.750 align:middle line:84% is not going to be equal to the complement of the intersection 00:51:13.750 --> 00:51:16.450 align:middle line:90% of the ci's. 00:51:16.450 --> 00:51:20.390 align:middle line:84% i equals 1 to infinity, which is, of course, 00:51:20.390 --> 00:51:22.105 align:middle line:90% the union of the Ui's. 00:51:22.105 --> 00:51:25.052 align:middle line:90% 00:51:25.052 --> 00:51:26.760 align:middle line:84% But this is a contradiction because we've 00:51:26.760 --> 00:51:29.220 align:middle line:84% assumed from the very beginning that the Ui's 00:51:29.220 --> 00:51:30.990 align:middle line:90% are an open cover of x. 00:51:30.990 --> 00:51:33.880 align:middle line:84% So what we've shown is that our open cover is not, 00:51:33.880 --> 00:51:37.480 align:middle line:84% in fact, an open cover, which is our contradiction. 00:51:37.480 --> 00:51:46.340 align:middle line:84% So we've shown so far that 1 implies 4, which implies 1, 00:51:46.340 --> 00:51:52.130 align:middle line:84% and 1 implies 2, which implies 1. 00:51:52.130 --> 00:51:56.600 align:middle line:84% So now, what we need to show is that 2 is the same as 3, i.e. 00:51:56.600 --> 00:51:59.150 align:middle line:84% sequentially compact is the same as totally bounded 00:51:59.150 --> 00:52:00.753 align:middle line:90% and Cauchy complete. 00:52:00.753 --> 00:52:03.170 align:middle line:84% Now, one of these directions has already been done for us. 00:52:03.170 --> 00:52:07.010 align:middle line:84% We've already shown that totally bounded is a consequence 00:52:07.010 --> 00:52:08.600 align:middle line:90% of sequentially compact. 00:52:08.600 --> 00:52:12.270 align:middle line:84% What we now need to show is that it's Cauchy complete. 00:52:12.270 --> 00:52:17.690 align:middle line:84% So we're going to first show that 2 implies 3. 00:52:17.690 --> 00:52:18.190 align:middle line:90% OK. 00:52:18.190 --> 00:52:22.610 align:middle line:90% 00:52:22.610 --> 00:52:40.190 align:middle line:84% So let xi be a Cauchy sequence of x, i.e. 00:52:40.190 --> 00:52:42.470 align:middle line:84% for all epsilon bigger than 0, there 00:52:42.470 --> 00:52:47.430 align:middle line:84% exists an n in the natural numbers such that for all n 00:52:47.430 --> 00:52:53.990 align:middle line:84% and m bigger than or equal to n, the distance between xn and xm 00:52:53.990 --> 00:52:55.670 align:middle line:90% is less than epsilon. 00:52:55.670 --> 00:52:58.800 align:middle line:84% We're going to use this to our advantage. 00:52:58.800 --> 00:53:03.320 align:middle line:84% So furthermore, what we know is that by sequential compactness, 00:53:03.320 --> 00:53:05.480 align:middle line:84% there's going to exist a convergent subsequence 00:53:05.480 --> 00:53:06.380 align:middle line:90% of the xi's. 00:53:06.380 --> 00:53:13.520 align:middle line:90% 00:53:13.520 --> 00:53:15.065 align:middle line:90% And let's say it converges to x. 00:53:15.065 --> 00:53:21.490 align:middle line:90% 00:53:21.490 --> 00:53:24.220 align:middle line:84% And what we're going to show then is that in fact, 00:53:24.220 --> 00:53:25.870 align:middle line:90% xi will converge to x. 00:53:25.870 --> 00:53:28.420 align:middle line:84% We're going to show that the limit point of the Cauchy 00:53:28.420 --> 00:53:33.370 align:middle line:84% sequence is, in fact, going to be in our metric space. 00:53:33.370 --> 00:53:35.560 align:middle line:84% To do that, let me move to-- actually, 00:53:35.560 --> 00:53:38.140 align:middle line:90% I can fit this in here. 00:53:38.140 --> 00:53:45.700 align:middle line:84% Notice then for all nk and n bigger than or equal to n, 00:53:45.700 --> 00:53:47.740 align:middle line:84% look at the distance from x to xn. 00:53:47.740 --> 00:53:50.410 align:middle line:84% And this will be less than or equal to the distance from x 00:53:50.410 --> 00:53:59.822 align:middle line:84% to xnk plus the distance from xnk to xn. 00:53:59.822 --> 00:54:01.280 align:middle line:84% And we know that this term is going 00:54:01.280 --> 00:54:04.012 align:middle line:90% to be less than epsilon over 2. 00:54:04.012 --> 00:54:05.720 align:middle line:84% Or sorry, we should assume epsilon over 2 00:54:05.720 --> 00:54:06.770 align:middle line:90% from the beginning. 00:54:06.770 --> 00:54:09.235 align:middle line:84% But furthermore, we know the xnk converges to x. 00:54:09.235 --> 00:54:11.360 align:middle line:84% So we can assume that this one is less than epsilon 00:54:11.360 --> 00:54:13.610 align:middle line:90% over 2, which is epsilon. 00:54:13.610 --> 00:54:17.100 align:middle line:90% So we've shown that xn is-- 00:54:17.100 --> 00:54:21.560 align:middle line:84% so what we've shown, therefore, is that xi will converge to x. 00:54:21.560 --> 00:54:26.340 align:middle line:84% And so we've shown that every Cauchy sequence is convergent. 00:54:26.340 --> 00:54:28.847 align:middle line:84% So we've shown Cauchy completeness, which 00:54:28.847 --> 00:54:29.930 align:middle line:90% is what we wanted to show. 00:54:29.930 --> 00:54:32.180 align:middle line:84% We wanted to show that sequential compactness was 00:54:32.180 --> 00:54:34.670 align:middle line:84% the same as totally bounded and Cauchy complete, 00:54:34.670 --> 00:54:36.830 align:middle line:84% which concludes the first part of our proof. 00:54:36.830 --> 00:54:39.370 align:middle line:90% 00:54:39.370 --> 00:54:41.440 align:middle line:84% The second part, the second direction, 00:54:41.440 --> 00:54:43.660 align:middle line:84% is going to be quite a bit harder because we 00:54:43.660 --> 00:54:47.320 align:middle line:84% want to show that every sequence has a convergent subsequence, 00:54:47.320 --> 00:54:49.540 align:middle line:84% assuming totally bounded and Cauchy complete, 00:54:49.540 --> 00:54:53.080 align:middle line:84% which is going to require quite a bit of work, 00:54:53.080 --> 00:54:56.050 align:middle line:84% or at least quite a bit of mental work, 00:54:56.050 --> 00:54:59.480 align:middle line:84% even if on the chalkboard it's not so much. 00:54:59.480 --> 00:55:01.270 align:middle line:84% It's a little bit confusing, how we 00:55:01.270 --> 00:55:07.080 align:middle line:84% go from totally bounded and Cauchy complete to a convergent 00:55:07.080 --> 00:55:09.264 align:middle line:90% subsequence. 00:55:09.264 --> 00:55:10.080 align:middle line:90% OK. 00:55:10.080 --> 00:55:14.250 align:middle line:84% We now want to show that property 4 implies property 1. 00:55:14.250 --> 00:55:15.645 align:middle line:84% And this will conclude our proof. 00:55:15.645 --> 00:55:21.860 align:middle line:90% 00:55:21.860 --> 00:55:22.360 align:middle line:90% OK. 00:55:22.360 --> 00:55:23.527 align:middle line:90% How are we going to do this? 00:55:23.527 --> 00:55:39.070 align:middle line:84% Well, going to assume that x is totally bounded 00:55:39.070 --> 00:55:40.015 align:middle line:90% and Cauchy complete. 00:55:40.015 --> 00:55:44.330 align:middle line:90% 00:55:44.330 --> 00:55:47.390 align:middle line:84% And we want to show sequential compactness. 00:55:47.390 --> 00:55:48.270 align:middle line:90% How do we do that? 00:55:48.270 --> 00:55:51.860 align:middle line:84% Well, we're going to start with an arbitrary sequence in x. 00:55:51.860 --> 00:56:02.318 align:middle line:84% So let xi be an arbitrary sequence in x. 00:56:02.318 --> 00:56:03.860 align:middle line:84% And we want to show that there exists 00:56:03.860 --> 00:56:05.570 align:middle line:84% a convergent subsequence, which is going 00:56:05.570 --> 00:56:07.670 align:middle line:90% to be a little bit difficult. 00:56:07.670 --> 00:56:08.630 align:middle line:90% OK. 00:56:08.630 --> 00:56:12.890 align:middle line:84% To do so, what we're going to do is start with totally bounded. 00:56:12.890 --> 00:56:20.490 align:middle line:84% We're going to note that for all n to the natural numbers, 00:56:20.490 --> 00:56:31.250 align:middle line:84% we have that there exists finitely many y1, or in fact, 00:56:31.250 --> 00:56:41.850 align:middle line:84% I'll notate this yn1 through yr of nn 00:56:41.850 --> 00:56:59.980 align:middle line:84% such that the balls of radius 1 over n around yi of n cover x. 00:56:59.980 --> 00:57:02.260 align:middle line:84% This is our assumption from totally boundedness. 00:57:02.260 --> 00:57:06.190 align:middle line:84% Here r of n is just a function of r. 00:57:06.190 --> 00:57:08.320 align:middle line:84% It's a function r that depends on n 00:57:08.320 --> 00:57:14.750 align:middle line:84% because the number of finitely many terms can depend on n. 00:57:14.750 --> 00:57:17.580 align:middle line:84% So in other words, we don't know exactly how many 00:57:17.580 --> 00:57:20.453 align:middle line:84% finitely many terms we have, but the more important thing 00:57:20.453 --> 00:57:22.370 align:middle line:84% to note is that we have finitely many of these 00:57:22.370 --> 00:57:26.510 align:middle line:84% for every single n, which is our superscript. 00:57:26.510 --> 00:57:27.890 align:middle line:90% OK. 00:57:27.890 --> 00:57:30.180 align:middle line:84% What are we going to do with this? 00:57:30.180 --> 00:57:34.418 align:middle line:84% Well, we're going to go from this to-- 00:57:34.418 --> 00:57:38.870 align:middle line:84% OK, we want to go from this to a Cauchy subsequence of the xi's. 00:57:38.870 --> 00:57:40.340 align:middle line:84% And once we know that it's Cauchy, 00:57:40.340 --> 00:57:43.650 align:middle line:84% we're going to know that it's convergent. 00:57:43.650 --> 00:57:51.160 align:middle line:84% So we know that there must exist 1-- 00:57:51.160 --> 00:57:56.317 align:middle line:90% 00:57:56.317 --> 00:57:57.400 align:middle line:90% how do I want to say this? 00:57:57.400 --> 00:58:05.050 align:middle line:90% 00:58:05.050 --> 00:58:09.070 align:middle line:84% Yeah, let's let S1 be the set of these points-- 00:58:09.070 --> 00:58:13.000 align:middle line:90% y1 n through yr of nn. 00:58:13.000 --> 00:58:16.150 align:middle line:90% 00:58:16.150 --> 00:58:19.030 align:middle line:84% What we know is that given that xi 00:58:19.030 --> 00:58:22.930 align:middle line:84% is an infinitely long sequence in x, 00:58:22.930 --> 00:58:28.210 align:middle line:84% one of the balls of radius 1 over n around these points-- 00:58:28.210 --> 00:58:31.240 align:middle line:90% 00:58:31.240 --> 00:58:33.250 align:middle line:90% sorry, I should call this Sn. 00:58:33.250 --> 00:58:36.970 align:middle line:84% We know that around the balls of radius 1 over n, around 1 00:58:36.970 --> 00:58:40.010 align:middle line:84% of these points, has to contain infinitely many of the terms. 00:58:40.010 --> 00:58:43.710 align:middle line:84% This is by the pigeonhole principle. 00:58:43.710 --> 00:58:47.220 align:middle line:84% So therefore, there exists in xi-- 00:58:47.220 --> 00:58:53.610 align:middle line:90% 00:58:53.610 --> 00:58:57.780 align:middle line:84% or sorry, I should say there exists a ball of radius 1 00:58:57.780 --> 00:59:11.530 align:middle line:84% around some yi of 1 containing infinitely many xi. 00:59:11.530 --> 00:59:14.130 align:middle line:90% 00:59:14.130 --> 00:59:16.350 align:middle line:84% And what we're going to do is choose our first point 00:59:16.350 --> 00:59:20.740 align:middle line:84% in our sequence to be one of these points. 00:59:20.740 --> 00:59:30.610 align:middle line:84% So let z1 be the first, just out of assumption. 00:59:30.610 --> 00:59:33.130 align:middle line:84% And what we're going to do is continue constructing these 00:59:33.130 --> 00:59:34.270 align:middle line:90% zi's. 00:59:34.270 --> 00:59:36.940 align:middle line:84% Well, we know then, furthermore, is 00:59:36.940 --> 00:59:46.210 align:middle line:84% that there exists a ball of radius 1 of yi 2 such that-- 00:59:46.210 --> 00:59:51.520 align:middle line:84% sorry, 1/2 such that the intersection of these 2 00:59:51.520 --> 00:59:58.360 align:middle line:84% ball of radius 1 over 2 around yi 2 00:59:58.360 --> 01:00:07.690 align:middle line:84% intersection the ball of radius 1 yj1 has infinitely many xi. 01:00:07.690 --> 01:00:11.620 align:middle line:90% 01:00:11.620 --> 01:00:13.680 align:middle line:84% How do we know that this is true? 01:00:13.680 --> 01:00:15.430 align:middle line:84% Well, we know that infinitely many of them 01:00:15.430 --> 01:00:17.020 align:middle line:90% already lie in this ball. 01:00:17.020 --> 01:00:19.510 align:middle line:84% And so we already know if we take the intersection of this 01:00:19.510 --> 01:00:22.120 align:middle line:84% with all the other balls of radius 1/2, 01:00:22.120 --> 01:00:24.083 align:middle line:84% there has to be infinitely many there. 01:00:24.083 --> 01:00:26.500 align:middle line:84% And to see this, I'll draw a little bit more of a picture. 01:00:26.500 --> 01:00:29.810 align:middle line:90% 01:00:29.810 --> 01:00:33.010 align:middle line:90% So here is our metric space x. 01:00:33.010 --> 01:00:36.400 align:middle line:84% And we start with covering it with balls of radius 1, 01:00:36.400 --> 01:00:40.060 align:middle line:84% which I'll draw pretty big, just so that we don't have to make 01:00:40.060 --> 01:00:41.720 align:middle line:90% a bunch of small pictures. 01:00:41.720 --> 01:00:43.270 align:middle line:84% So here are the balls of radius 1. 01:00:43.270 --> 01:00:48.290 align:middle line:90% 01:00:48.290 --> 01:00:51.530 align:middle line:84% Well, what we know then is that one of these-- 01:00:51.530 --> 01:00:54.020 align:middle line:90% let's say this one-- 01:00:54.020 --> 01:00:57.020 align:middle line:90% contains infinitely many terms. 01:00:57.020 --> 01:01:01.800 align:middle line:90% So we choose our z1 to be here. 01:01:01.800 --> 01:01:04.010 align:middle line:84% Furthermore, we have the intersection 01:01:04.010 --> 01:01:07.190 align:middle line:84% of this with balls of radius 1/2. 01:01:07.190 --> 01:01:10.290 align:middle line:84% So I make these a little bit smaller. 01:01:10.290 --> 01:01:13.310 align:middle line:90% Sorry, that's not quite good. 01:01:13.310 --> 01:01:15.490 align:middle line:84% Yeah, these will be my balls of radius 1/2. 01:01:15.490 --> 01:01:25.870 align:middle line:90% 01:01:25.870 --> 01:01:28.240 align:middle line:84% What I know then is that the intersection 01:01:28.240 --> 01:01:30.970 align:middle line:84% of the balls of radius 1/2 and the balls of radius 1-- 01:01:30.970 --> 01:01:33.010 align:middle line:84% there's only going to be finitely many of them, 01:01:33.010 --> 01:01:33.968 align:middle line:90% by assumption. 01:01:33.968 --> 01:01:36.010 align:middle line:84% And the fact that there are finitely many of them 01:01:36.010 --> 01:01:39.160 align:middle line:84% implies that there exists infinitely many of them 01:01:39.160 --> 01:01:41.050 align:middle line:90% that lie in one of these balls. 01:01:41.050 --> 01:01:43.150 align:middle line:84% One of these balls of radius 1/2 intersected 01:01:43.150 --> 01:01:46.270 align:middle line:90% one of the balls of radius 1. 01:01:46.270 --> 01:01:50.920 align:middle line:84% So what we can do is choose z2 to be 01:01:50.920 --> 01:01:52.580 align:middle line:90% in this intersection of them. 01:01:52.580 --> 01:01:54.340 align:middle line:84% And we're going to reiterate this process. 01:01:54.340 --> 01:01:56.543 align:middle line:84% We're going to assume that z3 is one 01:01:56.543 --> 01:01:58.960 align:middle line:84% of the infinitely many points in the intersection of balls 01:01:58.960 --> 01:02:03.800 align:middle line:84% of radius 1/3, 1/2, and 1, and so on and so forth. 01:02:03.800 --> 01:02:06.820 align:middle line:84% And what we hope to show is that the sequence 01:02:06.820 --> 01:02:08.650 align:middle line:90% is going to be Cauchy. 01:02:08.650 --> 01:02:13.187 align:middle line:84% But that's not, at this point, going to be too hard to do. 01:02:13.187 --> 01:02:14.770 align:middle line:84% OK, let me write this down more fully. 01:02:14.770 --> 01:02:19.570 align:middle line:84% We're going to choose zk to be in the intersection of balls 01:02:19.570 --> 01:02:25.135 align:middle line:84% of radius nk of n from n equals 1 to k. 01:02:25.135 --> 01:02:28.840 align:middle line:90% 01:02:28.840 --> 01:02:30.985 align:middle line:90% Yeah, balls of radius 1 to k. 01:02:30.985 --> 01:02:32.860 align:middle line:84% And now we want to show that this sequence is 01:02:32.860 --> 01:02:33.850 align:middle line:90% going to be Cauchy. 01:02:33.850 --> 01:02:36.550 align:middle line:90% Well, what's the distance-- 01:02:36.550 --> 01:02:42.380 align:middle line:84% or for all epsilon bigger than 0, 01:02:42.380 --> 01:02:46.850 align:middle line:84% choose n in the natural numbers such that 1 over n 01:02:46.850 --> 01:02:48.840 align:middle line:90% is less than epsilon. 01:02:48.840 --> 01:02:50.030 align:middle line:90% Then what do we know? 01:02:50.030 --> 01:02:54.800 align:middle line:84% Well, for all n and m bigger than or equal to n, 01:02:54.800 --> 01:03:00.260 align:middle line:84% we know then that zn and zm are both going to be contained 01:03:00.260 --> 01:03:04.033 align:middle line:84% in the ball of radius 1 over n of one of the yi's. 01:03:04.033 --> 01:03:06.200 align:middle line:84% I won't describe which one because it's a little bit 01:03:06.200 --> 01:03:06.920 align:middle line:90% annoying. 01:03:06.920 --> 01:03:09.620 align:middle line:84% But we know that they both must lie in 1 over n 01:03:09.620 --> 01:03:12.410 align:middle line:84% because they're in this intersection of all 01:03:12.410 --> 01:03:13.860 align:middle line:90% these terms. 01:03:13.860 --> 01:03:20.060 align:middle line:84% So what we know then is that the distance from zn to zm 01:03:20.060 --> 01:03:22.830 align:middle line:84% must be less than 1 over n or less than 01:03:22.830 --> 01:03:25.010 align:middle line:84% or equal to 1 over n, which is less 01:03:25.010 --> 01:03:28.610 align:middle line:84% than epsilon, which implies that the zi's must be Cauchy. 01:03:28.610 --> 01:03:33.920 align:middle line:84% So therefore, the zi's are Cauchy-- 01:03:33.920 --> 01:03:37.160 align:middle line:84% and by Cauchy completeness, we know that there 01:03:37.160 --> 01:03:40.180 align:middle line:90% exists a limit point. 01:03:40.180 --> 01:03:44.380 align:middle line:90% zi converges to some z in x. 01:03:44.380 --> 01:03:47.140 align:middle line:84% But recall again that our zi's are constructed 01:03:47.140 --> 01:03:49.300 align:middle line:90% from our original sequence. 01:03:49.300 --> 01:03:52.190 align:middle line:84% They're a subsequence of our original sequence. 01:03:52.190 --> 01:03:53.800 align:middle line:84% So what we've shown is that there 01:03:53.800 --> 01:03:57.020 align:middle line:84% is a convergent subsequence of our sequence in x. 01:03:57.020 --> 01:03:59.470 align:middle line:84% And this is exactly what we sought out to show. 01:03:59.470 --> 01:04:03.100 align:middle line:84% And this shows that totally bounded and Cauchy complete 01:04:03.100 --> 01:04:05.260 align:middle line:90% implies sequentially compact. 01:04:05.260 --> 01:04:07.840 align:middle line:84% And this is the end of our four-part proof. 01:04:07.840 --> 01:04:10.520 align:middle line:90% 01:04:10.520 --> 01:04:13.340 align:middle line:84% So again, to reiterate, we've shown 01:04:13.340 --> 01:04:16.010 align:middle line:84% that topologically compact is the same as sequentially 01:04:16.010 --> 01:04:18.860 align:middle line:84% compact, where here we used the Lebesgue number lemma 01:04:18.860 --> 01:04:21.290 align:middle line:90% and totally boundedness. 01:04:21.290 --> 01:04:23.720 align:middle line:84% Then from here we show that sequentially compact 01:04:23.720 --> 01:04:26.472 align:middle line:84% is the same as totally bounded and Cauchy complete. 01:04:26.472 --> 01:04:28.430 align:middle line:84% Cauchy complete didn't take too much more work. 01:04:28.430 --> 01:04:29.700 align:middle line:90% That was pretty short. 01:04:29.700 --> 01:04:32.960 align:middle line:84% The other direction required quite a bit of maneuvering 01:04:32.960 --> 01:04:34.790 align:middle line:84% to get that convergent subsequence, 01:04:34.790 --> 01:04:38.002 align:middle line:84% but again, the notes contain this more fully 01:04:38.002 --> 01:04:40.460 align:middle line:84% if you want to read through it slowly and at your own pace. 01:04:40.460 --> 01:04:43.090 align:middle line:90% 01:04:43.090 --> 01:04:46.480 align:middle line:84% And furthermore, lastly, we showed that 1 is the same as 4, 01:04:46.480 --> 01:04:49.540 align:middle line:84% where topologically compact is the same as closed subsets 01:04:49.540 --> 01:04:51.790 align:middle line:84% with the finite intersection property having 01:04:51.790 --> 01:04:53.328 align:middle line:90% nonempty intersection. 01:04:53.328 --> 01:04:55.870 align:middle line:84% And now, let's think carefully about how each of these proofs 01:04:55.870 --> 01:04:58.090 align:middle line:84% went because each have their own purpose 01:04:58.090 --> 01:05:02.650 align:middle line:90% in a proof about metric spaces. 01:05:02.650 --> 01:05:06.320 align:middle line:84% To show that 1 was the same as 4, we looked at open covers, 01:05:06.320 --> 01:05:06.820 align:middle line:90% right? 01:05:06.820 --> 01:05:08.830 align:middle line:84% We took the open cover definition 01:05:08.830 --> 01:05:12.010 align:middle line:84% and looked at complements of closed sets and being open, 01:05:12.010 --> 01:05:13.780 align:middle line:84% and complements of open sets being closed. 01:05:13.780 --> 01:05:15.988 align:middle line:84% And we used all these different properties like that. 01:05:15.988 --> 01:05:18.880 align:middle line:84% And that makes a lot of sense because both definitions 01:05:18.880 --> 01:05:21.760 align:middle line:84% are in terms of closed and open sets. 01:05:21.760 --> 01:05:24.050 align:middle line:84% But to show sequentially compact was 01:05:24.050 --> 01:05:26.050 align:middle line:84% the same as totally bounded and Cauchy complete, 01:05:26.050 --> 01:05:30.400 align:middle line:84% we used definitions of sequential compactness, right? 01:05:30.400 --> 01:05:32.800 align:middle line:84% Because it's so much easier to look 01:05:32.800 --> 01:05:34.630 align:middle line:84% at convergent subsequences and such 01:05:34.630 --> 01:05:37.120 align:middle line:84% when you're looking at sequential compactness. 01:05:37.120 --> 01:05:39.983 align:middle line:84% So we could have proven that 1 implies 3, 01:05:39.983 --> 01:05:41.650 align:middle line:84% but it would have been a little bit more 01:05:41.650 --> 01:05:44.360 align:middle line:84% difficult and a little bit harder to maneuver around. 01:05:44.360 --> 01:05:47.320 align:middle line:84% And in fact, that's why I've shown these four properties 01:05:47.320 --> 01:05:50.710 align:middle line:84% today because if you're in a class where you're proving 01:05:50.710 --> 01:05:54.830 align:middle line:84% facts about metric spaces, you need 01:05:54.830 --> 01:05:57.470 align:middle line:84% to use the property that makes the most 01:05:57.470 --> 01:05:59.120 align:middle line:90% sense to use in a given moment. 01:05:59.120 --> 01:06:00.620 align:middle line:84% For instance, on your problem set, 01:06:00.620 --> 01:06:03.020 align:middle line:84% you're going to be asked to show that the union 01:06:03.020 --> 01:06:06.870 align:middle line:84% of finitely many compact sets is going to be compact. 01:06:06.870 --> 01:06:09.650 align:middle line:84% How do you know that this is going to be true? 01:06:09.650 --> 01:06:11.710 align:middle line:84% Well, you can use sequential compactness. 01:06:11.710 --> 01:06:13.265 align:middle line:84% You can use topological compactness. 01:06:13.265 --> 01:06:14.890 align:middle line:84% You can use any of these for properties 01:06:14.890 --> 01:06:17.740 align:middle line:84% that you still wish to choose, but be sure to think carefully 01:06:17.740 --> 01:06:19.390 align:middle line:84% about which one because one of them 01:06:19.390 --> 01:06:20.690 align:middle line:90% might be a little bit easier. 01:06:20.690 --> 01:06:24.350 align:middle line:84% So I won't discredit you if you use any of the other ones. 01:06:24.350 --> 01:06:26.080 align:middle line:84% And this is true in general when you're 01:06:26.080 --> 01:06:28.100 align:middle line:84% going into proofs about metric spaces. 01:06:28.100 --> 01:06:30.100 align:middle line:84% Having the intuition to flip from one definition 01:06:30.100 --> 01:06:34.820 align:middle line:84% to the other is deeply important and very useful to know. 01:06:34.820 --> 01:06:37.300 align:middle line:84% So that will conclude today's lecture. 01:06:37.300 --> 01:06:41.200 align:middle line:84% On next Thursday, we're going to prove some new facts 01:06:41.200 --> 01:06:42.430 align:middle line:90% about compact spaces. 01:06:42.430 --> 01:06:44.350 align:middle line:84% And in particular, what we're going 01:06:44.350 --> 01:06:47.080 align:middle line:84% to show or talk about is the history 01:06:47.080 --> 01:06:49.390 align:middle line:84% of compact metric spaces and why they were developed 01:06:49.390 --> 01:06:51.250 align:middle line:84% in the first place, which comes from what 01:06:51.250 --> 01:06:53.140 align:middle line:84% is known as the Dirichlet problem, 01:06:53.140 --> 01:06:55.570 align:middle line:84% but I'll leave that discussion until next Thursday. 01:06:55.570 --> 01:06:57.890 align:middle line:90% Until then have a great weekend. 01:06:57.890 --> 01:06:59.670 align:middle line:90% Thank you. 01:06:59.670 --> 01:07:05.000 align:middle line:90%