WEBVTT 00:00:00.000 --> 00:00:02.440 align:middle line:90% [SQUEAKING] 00:00:02.440 --> 00:00:04.392 align:middle line:90% [RUSTLING] 00:00:04.392 --> 00:00:07.808 align:middle line:90% [CLICKING] 00:00:07.808 --> 00:00:14.460 align:middle line:90% 00:00:14.460 --> 00:00:16.560 align:middle line:84% TOBIAS COLDING: OK, so last time we 00:00:16.560 --> 00:00:19.300 align:middle line:90% were looking at a metric space. 00:00:19.300 --> 00:00:20.470 align:middle line:90% So this is a metric space. 00:00:20.470 --> 00:00:25.840 align:middle line:90% 00:00:25.840 --> 00:00:30.830 align:middle line:84% And we looked at what we called a open subset. 00:00:30.830 --> 00:00:37.920 align:middle line:90% 00:00:37.920 --> 00:00:42.240 align:middle line:84% And so an open subset is-- so over here 00:00:42.240 --> 00:00:59.710 align:middle line:84% is now a subset of x is said to be open if for all x in O 00:00:59.710 --> 00:01:12.030 align:middle line:84% there exists r greater than 0 so that the entire ball around x 00:01:12.030 --> 00:01:14.470 align:middle line:90% is contained in O. 00:01:14.470 --> 00:01:20.510 align:middle line:84% And so the picture is that here you have x. 00:01:20.510 --> 00:01:23.950 align:middle line:84% Here you have some open subset O. 00:01:23.950 --> 00:01:29.710 align:middle line:84% And so if for each point x in O, there 00:01:29.710 --> 00:01:36.150 align:middle line:84% is an entire ball here that is contained in O, 00:01:36.150 --> 00:01:37.960 align:middle line:90% then O is said to be open. 00:01:37.960 --> 00:01:42.470 align:middle line:90% 00:01:42.470 --> 00:01:46.970 align:middle line:84% One of a few things that we proved is-- 00:01:46.970 --> 00:01:51.873 align:middle line:90% so one thing we proved is that-- 00:01:51.873 --> 00:01:54.910 align:middle line:90% 00:01:54.910 --> 00:01:56.850 align:middle line:84% so again, x, d is a metric space. 00:01:56.850 --> 00:01:59.400 align:middle line:90% 00:01:59.400 --> 00:02:08.380 align:middle line:84% Sorry, I should say we defined the ball, Br of x. 00:02:08.380 --> 00:02:14.060 align:middle line:90% So here, x is in x. 00:02:14.060 --> 00:02:16.460 align:middle line:90% r is some positive number. 00:02:16.460 --> 00:02:23.260 align:middle line:84% And then this thing here is then the y in x such 00:02:23.260 --> 00:02:31.740 align:middle line:84% that the distance from x to y is strictly less than r. 00:02:31.740 --> 00:02:41.410 align:middle line:84% And we showed that Br of x is given. 00:02:41.410 --> 00:02:44.020 align:middle line:90% 00:02:44.020 --> 00:02:47.780 align:middle line:84% By that Br, we say also that Br of x 00:02:47.780 --> 00:02:56.750 align:middle line:84% is a ball of radius r centered at x. 00:02:56.750 --> 00:03:05.570 align:middle line:90% 00:03:05.570 --> 00:03:10.390 align:middle line:84% Now, there's a few other things that we discussed. 00:03:10.390 --> 00:03:13.650 align:middle line:90% 00:03:13.650 --> 00:03:24.330 align:middle line:84% So one thing is that so on subsets, 00:03:24.330 --> 00:03:26.150 align:middle line:90% we have three operations. 00:03:26.150 --> 00:03:37.610 align:middle line:90% 00:03:37.610 --> 00:03:39.310 align:middle line:90% One is intersection. 00:03:39.310 --> 00:03:44.570 align:middle line:90% 00:03:44.570 --> 00:04:00.770 align:middle line:84% So this is that if u alpha is a family of subsets, then 00:04:00.770 --> 00:04:05.570 align:middle line:84% we look at the intersection of all these subsets. 00:04:05.570 --> 00:04:10.290 align:middle line:84% And so x is in this intersection just 00:04:10.290 --> 00:04:16.720 align:middle line:84% means that x is in u alpha for all alpha. 00:04:16.720 --> 00:04:21.209 align:middle line:90% 00:04:21.209 --> 00:04:27.010 align:middle line:84% The second operation we looked at was union. 00:04:27.010 --> 00:04:31.570 align:middle line:84% And so again, u alpha is a family of subsets. 00:04:31.570 --> 00:04:38.210 align:middle line:84% Then we say that x here is in the union of the u alpha. 00:04:38.210 --> 00:04:50.370 align:middle line:84% This is the same as saying that there exists alpha such that x 00:04:50.370 --> 00:04:53.640 align:middle line:90% is in u alpha. 00:04:53.640 --> 00:04:56.440 align:middle line:90% And the third thing we-- 00:04:56.440 --> 00:05:01.920 align:middle line:84% the celebration we have is that if you have a subset capital 00:05:01.920 --> 00:05:07.500 align:middle line:90% U, then we have the complement. 00:05:07.500 --> 00:05:21.760 align:middle line:84% So the complement of U is the x such 00:05:21.760 --> 00:05:27.860 align:middle line:84% that x is not in U. That's a complement. 00:05:27.860 --> 00:05:33.360 align:middle line:84% And so there was these three operations on subsets. 00:05:33.360 --> 00:05:37.280 align:middle line:90% And why this is relevant here? 00:05:37.280 --> 00:05:43.360 align:middle line:84% This is relevant because if u alpha-- 00:05:43.360 --> 00:05:46.150 align:middle line:90% so we have the following lemma. 00:05:46.150 --> 00:05:51.470 align:middle line:90% 00:05:51.470 --> 00:05:53.710 align:middle line:84% Again, we have a metric space here, 00:05:53.710 --> 00:06:10.630 align:middle line:84% and we have O alpha is a family of open subset of x. 00:06:10.630 --> 00:06:17.640 align:middle line:90% Then the union is also open. 00:06:17.640 --> 00:06:21.310 align:middle line:90% 00:06:21.310 --> 00:06:23.920 align:middle line:90% Now why is this the case? 00:06:23.920 --> 00:06:28.070 align:middle line:90% 00:06:28.070 --> 00:06:35.470 align:middle line:84% Well, it's the case because if x here 00:06:35.470 --> 00:06:40.630 align:middle line:84% is in the union of the O alpha, then 00:06:40.630 --> 00:06:50.380 align:middle line:84% this means that there exists alpha such that x here 00:06:50.380 --> 00:06:53.340 align:middle line:90% is in alpha. 00:06:53.340 --> 00:06:58.860 align:middle line:84% But since that alpha is open, this means that there exists-- 00:06:58.860 --> 00:07:01.060 align:middle line:84% so not only is x in that, but there 00:07:01.060 --> 00:07:14.060 align:middle line:84% exists r that depends on x and alpha r positive, 00:07:14.060 --> 00:07:22.380 align:middle line:84% so that the whole ball centered at x is contained in O alpha. 00:07:22.380 --> 00:07:26.210 align:middle line:84% But if it's contained in alpha then this imply-- 00:07:26.210 --> 00:07:32.180 align:middle line:90% 00:07:32.180 --> 00:07:39.420 align:middle line:84% if it's contained in alpha, then this 00:07:39.420 --> 00:07:42.100 align:middle line:84% implies that it's, of course, this ball 00:07:42.100 --> 00:07:49.960 align:middle line:84% here is contained in the union of these guys. 00:07:49.960 --> 00:07:53.450 align:middle line:90% 00:07:53.450 --> 00:07:57.490 align:middle line:84% But that's exactly what it means, 00:07:57.490 --> 00:07:59.620 align:middle line:90% to say that the union is open. 00:07:59.620 --> 00:08:07.890 align:middle line:90% 00:08:07.890 --> 00:08:13.810 align:middle line:84% The other lemma related to this is-- 00:08:13.810 --> 00:08:16.890 align:middle line:84% and there, you have to be careful 00:08:16.890 --> 00:08:20.270 align:middle line:90% that if you have finitely many. 00:08:20.270 --> 00:08:28.290 align:middle line:84% So can you have a metric space and O1 up 00:08:28.290 --> 00:08:37.540 align:middle line:84% to On finitely many open subsets? 00:08:37.540 --> 00:08:43.650 align:middle line:90% 00:08:43.650 --> 00:08:48.160 align:middle line:84% If you have finitely many open subsets, 00:08:48.160 --> 00:08:54.500 align:middle line:84% then the intersection of these is also open. 00:08:54.500 --> 00:09:00.520 align:middle line:90% 00:09:00.520 --> 00:09:03.560 align:middle line:90% Now why is that the case? 00:09:03.560 --> 00:09:09.360 align:middle line:84% This is the case because if you have-- 00:09:09.360 --> 00:09:18.760 align:middle line:84% so if x here is in the intersection of these guys, 00:09:18.760 --> 00:09:23.480 align:middle line:84% then it means that this is equivalent to saying that x here 00:09:23.480 --> 00:09:26.180 align:middle line:90% is in z for all i-- 00:09:26.180 --> 00:09:28.960 align:middle line:90% 00:09:28.960 --> 00:09:30.860 align:middle line:90% i equal to 1 up to n. 00:09:30.860 --> 00:09:37.960 align:middle line:90% 00:09:37.960 --> 00:09:48.510 align:middle line:84% But since Oi is open, then for Oi, there exists ri. 00:09:48.510 --> 00:09:59.230 align:middle line:84% This means that since Oi is open, there exists at ri such 00:09:59.230 --> 00:10:11.790 align:middle line:84% that the ball here with radius r but center x, this one here 00:10:11.790 --> 00:10:12.990 align:middle line:90% is contained in Oi. 00:10:12.990 --> 00:10:15.590 align:middle line:90% 00:10:15.590 --> 00:10:17.950 align:middle line:90% This is for each i. 00:10:17.950 --> 00:10:20.750 align:middle line:84% But now if we check-- there's only finitely many 00:10:20.750 --> 00:10:27.040 align:middle line:84% i so you can let r be the minimum of these guys. 00:10:27.040 --> 00:10:30.230 align:middle line:90% 00:10:30.230 --> 00:10:35.710 align:middle line:84% If r is the minimum of these guys, then 00:10:35.710 --> 00:10:38.530 align:middle line:90% the ball of radius r around x-- 00:10:38.530 --> 00:10:42.710 align:middle line:84% these are the points where the distance is strictly less than r 00:10:42.710 --> 00:10:44.260 align:middle line:90% to x. 00:10:44.260 --> 00:10:50.620 align:middle line:84% But that is, of course, contained in Br i of x. 00:10:50.620 --> 00:10:53.540 align:middle line:84% This is for each i, you have this. 00:10:53.540 --> 00:10:59.020 align:middle line:84% But this one here was supposed to be contained in Oi. 00:10:59.020 --> 00:11:04.700 align:middle line:84% And so you see that this ball here-- 00:11:04.700 --> 00:11:15.460 align:middle line:84% again, this ball here is contained in this for each i. 00:11:15.460 --> 00:11:17.500 align:middle line:84% So this means that this ball must 00:11:17.500 --> 00:11:22.130 align:middle line:84% be contained in the intersection of these guys. 00:11:22.130 --> 00:11:26.020 align:middle line:90% 00:11:26.020 --> 00:11:28.960 align:middle line:84% And I'll just write it out again, 00:11:28.960 --> 00:11:36.940 align:middle line:84% just emphasizing that I'm only doing this for finitely many Os. 00:11:36.940 --> 00:11:39.500 align:middle line:84% And that's exactly what it means, 00:11:39.500 --> 00:11:41.860 align:middle line:90% that the intersection is open. 00:11:41.860 --> 00:11:45.580 align:middle line:84% And again, you could draw a little picture here. 00:11:45.580 --> 00:11:48.100 align:middle line:84% It might be helpful to draw a little picture. 00:11:48.100 --> 00:11:52.860 align:middle line:90% You have this here is O1. 00:11:52.860 --> 00:11:55.700 align:middle line:90% Here, you have O2. 00:11:55.700 --> 00:11:58.220 align:middle line:90% And here, you have O3. 00:11:58.220 --> 00:12:01.380 align:middle line:84% You have a point in the intersection. 00:12:01.380 --> 00:12:04.940 align:middle line:84% There's a ball here that is contained in this. 00:12:04.940 --> 00:12:09.100 align:middle line:84% There's another ball here that's contained in this. 00:12:09.100 --> 00:12:11.940 align:middle line:84% And there's a third ball, this ball here. 00:12:11.940 --> 00:12:14.500 align:middle line:90% So that's contained in that. 00:12:14.500 --> 00:12:17.860 align:middle line:84% And then you look at the smallest ball. 00:12:17.860 --> 00:12:21.340 align:middle line:84% You look at the smallest weight-- the minimum, again, 00:12:21.340 --> 00:12:23.780 align:middle line:90% of the radii. 00:12:23.780 --> 00:12:30.180 align:middle line:84% And that ball is then contained in the intersection of the 3. 00:12:30.180 --> 00:12:36.980 align:middle line:90% 00:12:36.980 --> 00:12:39.080 align:middle line:84% Now why do we have to be careful? 00:12:39.080 --> 00:12:43.090 align:middle line:84% Let me just remind you, why do we have to be careful here? 00:12:43.090 --> 00:12:47.930 align:middle line:84% And so the examples that we looked at last time 00:12:47.930 --> 00:13:00.650 align:middle line:90% was that if you are on-- 00:13:00.650 --> 00:13:07.490 align:middle line:84% so the example from last time is that if x here 00:13:07.490 --> 00:13:14.026 align:middle line:84% is the real number and you're looking at On-- 00:13:14.026 --> 00:13:18.610 align:middle line:90% n here is a natural number-- 00:13:18.610 --> 00:13:25.010 align:middle line:84% and you're looking at this interval here from minus 1 00:13:25.010 --> 00:13:30.010 align:middle line:84% over n to n, well, then if you're 00:13:30.010 --> 00:13:36.150 align:middle line:84% looking at the intersection of all of these, 00:13:36.150 --> 00:13:39.470 align:middle line:90% well, that's just the one point. 00:13:39.470 --> 00:13:42.640 align:middle line:90% 00:13:42.640 --> 00:13:46.220 align:middle line:84% And of course, 0 is not an open subset. 00:13:46.220 --> 00:13:49.200 align:middle line:84% Because if it was an open subset, a whole ball-- 00:13:49.200 --> 00:13:52.680 align:middle line:84% and a ball here on the line is like a little open 00:13:52.680 --> 00:13:55.620 align:middle line:84% little interval that doesn't have the two endpoints. 00:13:55.620 --> 00:13:58.180 align:middle line:90% 00:13:58.180 --> 00:14:11.520 align:middle line:84% So note that in this example, each Oi or each On is open, 00:14:11.520 --> 00:14:20.670 align:middle line:84% but the intersection of the On's is not open. 00:14:20.670 --> 00:14:23.880 align:middle line:90% 00:14:23.880 --> 00:14:28.200 align:middle line:84% And the problem, again, is that there are infinitely many 00:14:28.200 --> 00:14:28.760 align:middle line:90% of n's. 00:14:28.760 --> 00:14:32.000 align:middle line:90% 00:14:32.000 --> 00:14:37.070 align:middle line:84% Now again, one of the things we proved was-- 00:14:37.070 --> 00:14:42.470 align:middle line:84% and maybe I was just here on that board-- 00:14:42.470 --> 00:14:45.670 align:middle line:84% but one of the things we proved last time 00:14:45.670 --> 00:14:53.920 align:middle line:84% was that if you take a ball, then it is actually an open set. 00:14:53.920 --> 00:14:58.910 align:middle line:90% 00:14:58.910 --> 00:15:10.230 align:middle line:84% Let me prove a variation of that. 00:15:10.230 --> 00:15:16.030 align:middle line:84% So suppose you're looking at-- so let's have a lemma here 00:15:16.030 --> 00:15:19.830 align:middle line:84% that if you have, again, a metric space here 00:15:19.830 --> 00:15:22.830 align:middle line:90% and r here is bigger than 0. 00:15:22.830 --> 00:15:28.230 align:middle line:84% Now let's look at the x and then take away. 00:15:28.230 --> 00:15:30.490 align:middle line:90% But now I'm looking at the sets. 00:15:30.490 --> 00:15:34.030 align:middle line:84% So let me look at the set y here. 00:15:34.030 --> 00:15:41.300 align:middle line:84% So x is some fixed point in this metric space, 00:15:41.300 --> 00:15:52.340 align:middle line:84% and you're looking at the y in x so that the distance to x from y 00:15:52.340 --> 00:15:55.300 align:middle line:90% is strictly bigger than r. 00:15:55.300 --> 00:16:06.300 align:middle line:84% So in other words, if this here is x, this is the space x. 00:16:06.300 --> 00:16:08.540 align:middle line:90% Here, you have the point x. 00:16:08.540 --> 00:16:14.060 align:middle line:84% Then you have here this, including this circle here, 00:16:14.060 --> 00:16:17.820 align:middle line:84% are the points that have distance at most r 00:16:17.820 --> 00:16:20.500 align:middle line:90% from little x. 00:16:20.500 --> 00:16:27.140 align:middle line:84% Then this is the set here outside this disk, 00:16:27.140 --> 00:16:32.740 align:middle line:84% where the boundary is included, so outside. 00:16:32.740 --> 00:16:37.210 align:middle line:84% And so what I want to prove here is that this here 00:16:37.210 --> 00:16:40.640 align:middle line:90% is this here is an open subset. 00:16:40.640 --> 00:16:44.810 align:middle line:90% 00:16:44.810 --> 00:16:48.170 align:middle line:84% And the proof is very similar to how 00:16:48.170 --> 00:16:50.910 align:middle line:90% we proved that a ball is open. 00:16:50.910 --> 00:16:59.290 align:middle line:84% Again, the key ingredient is the triangle inequality. 00:16:59.290 --> 00:17:12.410 align:middle line:90% And so let's try to prove that. 00:17:12.410 --> 00:17:26.710 align:middle line:84% So we take y here in this-- let's call this set something. 00:17:26.710 --> 00:17:32.280 align:middle line:90% Let's call it E. 00:17:32.280 --> 00:17:36.960 align:middle line:84% So suppose you have x here, y here in E. 00:17:36.960 --> 00:17:41.720 align:middle line:84% So this means that the distance from y to x 00:17:41.720 --> 00:17:44.760 align:middle line:90% is strictly bigger than r. 00:17:44.760 --> 00:17:47.260 align:middle line:90% I want to prove-- 00:17:47.260 --> 00:17:51.720 align:middle line:90% so I have my picture here. 00:17:51.720 --> 00:17:52.700 align:middle line:90% This is x. 00:17:52.700 --> 00:17:54.680 align:middle line:90% This is this ball here. 00:17:54.680 --> 00:17:58.080 align:middle line:84% And then I have a y that lie outside it. 00:17:58.080 --> 00:18:01.160 align:middle line:84% And I want to find a little ball here 00:18:01.160 --> 00:18:04.720 align:middle line:84% around y that is entirely contained 00:18:04.720 --> 00:18:07.400 align:middle line:90% in the complement of this ball. 00:18:07.400 --> 00:18:12.920 align:middle line:84% And so I do the same that I did last time-- 00:18:12.920 --> 00:18:16.280 align:middle line:84% I mean, almost the same, not exactly the same. 00:18:16.280 --> 00:18:19.500 align:middle line:90% So we said S here equal to y. 00:18:19.500 --> 00:18:23.200 align:middle line:84% So the idea is that you're looking at this distance here. 00:18:23.200 --> 00:18:27.240 align:middle line:90% This distance is d x, y. 00:18:27.240 --> 00:18:32.040 align:middle line:90% This part of it is r. 00:18:32.040 --> 00:18:38.160 align:middle line:84% And so it seems reasonable, this here, if this picture was right, 00:18:38.160 --> 00:18:42.720 align:middle line:90% then this thing here would be s. 00:18:42.720 --> 00:18:50.200 align:middle line:84% And this s here would be d x, y minus r. 00:18:50.200 --> 00:18:54.580 align:middle line:84% So that's what I'm setting s equal to. 00:18:54.580 --> 00:18:58.000 align:middle line:90% 00:18:58.000 --> 00:19:00.920 align:middle line:84% And the first thing I noticed that because this is strictly 00:19:00.920 --> 00:19:06.400 align:middle line:84% bigger than r, this thing here is strictly positive. 00:19:06.400 --> 00:19:11.840 align:middle line:84% And then I take set here in that ball. 00:19:11.840 --> 00:19:13.840 align:middle line:90% Now it's centered at y. 00:19:13.840 --> 00:19:16.880 align:middle line:90% Let's have radius s. 00:19:16.880 --> 00:19:22.240 align:middle line:84% And then I want to look at the distance to-- 00:19:22.240 --> 00:19:23.860 align:middle line:90% and I want to estimate. 00:19:23.860 --> 00:19:44.470 align:middle line:84% So I want to estimate from below the distance from x to z. 00:19:44.470 --> 00:19:46.375 align:middle line:84% Well, the triangle inequality, if you just 00:19:46.375 --> 00:19:47.750 align:middle line:84% think about the triangle-- that's 00:19:47.750 --> 00:19:51.250 align:middle line:84% something like bounding a distance from above. 00:19:51.250 --> 00:19:54.150 align:middle line:84% So if I want to estimate something below, 00:19:54.150 --> 00:19:58.430 align:middle line:84% then basically, this thing I want to estimate from below 00:19:58.430 --> 00:20:01.550 align:middle line:84% have to be on the other side of the inequality. 00:20:01.550 --> 00:20:05.450 align:middle line:90% And so how does that work? 00:20:05.450 --> 00:20:09.230 align:middle line:90% 00:20:09.230 --> 00:20:14.390 align:middle line:84% So let's again draw a little picture, too. 00:20:14.390 --> 00:20:17.870 align:middle line:84% So let me do a little bit on the side. 00:20:17.870 --> 00:20:24.270 align:middle line:84% So I have this ball here, and I have this y here, 00:20:24.270 --> 00:20:26.010 align:middle line:90% and then I have this z here. 00:20:26.010 --> 00:20:30.180 align:middle line:90% 00:20:30.180 --> 00:20:32.540 align:middle line:90% And I now want to say-- 00:20:32.540 --> 00:20:35.980 align:middle line:90% I want to estimate. 00:20:35.980 --> 00:20:42.220 align:middle line:84% So I'm saying that the distance from-- 00:20:42.220 --> 00:20:48.980 align:middle line:84% so this is what I want to bound from below. 00:20:48.980 --> 00:20:54.340 align:middle line:84% And what I know is that this set here is-- 00:20:54.340 --> 00:20:59.020 align:middle line:84% I know something about the distance to y. 00:20:59.020 --> 00:21:02.420 align:middle line:90% So this here, I know. 00:21:02.420 --> 00:21:06.820 align:middle line:84% And so you see, this thing here by the triangle inequality 00:21:06.820 --> 00:21:15.220 align:middle line:84% is now bounded by the distance from x to y. 00:21:15.220 --> 00:21:20.540 align:middle line:84% And I know that this thing here is 00:21:20.540 --> 00:21:24.380 align:middle line:84% less than or equal to this here is the one I want to estimate. 00:21:24.380 --> 00:21:26.010 align:middle line:90% So I'm just leaving that. 00:21:26.010 --> 00:21:29.970 align:middle line:84% This one here, it's because set is in the ball of radius 00:21:29.970 --> 00:21:31.530 align:middle line:90% s around y. 00:21:31.530 --> 00:21:39.690 align:middle line:84% So this thing here is actually strictly less than s, like that. 00:21:39.690 --> 00:21:43.090 align:middle line:90% 00:21:43.090 --> 00:21:46.050 align:middle line:84% And so now you can just move this one here over 00:21:46.050 --> 00:21:47.450 align:middle line:90% on the other side. 00:21:47.450 --> 00:21:58.010 align:middle line:84% And you have that d x, y minus s is strictly less than d x 00:21:58.010 --> 00:21:59.930 align:middle line:90% comma z. 00:21:59.930 --> 00:22:03.290 align:middle line:84% Now we have to fill in what S is. 00:22:03.290 --> 00:22:05.370 align:middle line:90% S is up there. 00:22:05.370 --> 00:22:10.010 align:middle line:84% So this thing here is-- this thing here 00:22:10.010 --> 00:22:12.710 align:middle line:90% is equal to this is dx. 00:22:12.710 --> 00:22:14.590 align:middle line:90% I'm just copying that. 00:22:14.590 --> 00:22:21.710 align:middle line:84% But I fill in what s is, and s is d x, y minus r. 00:22:21.710 --> 00:22:25.360 align:middle line:90% 00:22:25.360 --> 00:22:29.080 align:middle line:84% But you see, those two cancel, and you're left with the r. 00:22:29.080 --> 00:22:31.320 align:middle line:84% And so you have that r is strictly 00:22:31.320 --> 00:22:35.440 align:middle line:84% less than this, which is exactly what we wanted to prove. 00:22:35.440 --> 00:22:41.640 align:middle line:84% And so, again, it was almost exactly the same proof 00:22:41.640 --> 00:22:44.180 align:middle line:84% that we had to prove that the ball was open. 00:22:44.180 --> 00:22:49.360 align:middle line:90% 00:22:49.360 --> 00:22:51.760 align:middle line:84% What we're using here is that, again, we're 00:22:51.760 --> 00:22:54.720 align:middle line:84% using that the triangle inequality is 00:22:54.720 --> 00:22:57.800 align:middle line:84% like you're bounding some distance from above. 00:22:57.800 --> 00:23:02.800 align:middle line:84% And so you have to use it the other way with having the things 00:23:02.800 --> 00:23:05.800 align:middle line:84% that you want to bound from below on the other side 00:23:05.800 --> 00:23:06.980 align:middle line:90% of the inequality. 00:23:06.980 --> 00:23:09.720 align:middle line:90% 00:23:09.720 --> 00:23:14.160 align:middle line:84% So the next thing is then-- and we talked a little bit about 00:23:14.160 --> 00:23:16.160 align:middle line:90% that last time-- 00:23:16.160 --> 00:23:19.480 align:middle line:84% the next thing is closed subsets. 00:23:19.480 --> 00:23:23.150 align:middle line:84% So this may seem like a triviality 00:23:23.150 --> 00:23:29.110 align:middle line:84% because it's defined, in a way, trivially. 00:23:29.110 --> 00:23:30.150 align:middle line:90% So what is it? 00:23:30.150 --> 00:23:32.960 align:middle line:84% But it's actually a really useful notion. 00:23:32.960 --> 00:23:37.390 align:middle line:90% 00:23:37.390 --> 00:23:38.650 align:middle line:90% And so this is-- 00:23:38.650 --> 00:23:42.670 align:middle line:90% 00:23:42.670 --> 00:23:50.890 align:middle line:84% so a subset C is said to be closed-- 00:23:50.890 --> 00:23:52.590 align:middle line:90% again, x is a metric space-- 00:23:52.590 --> 00:24:04.970 align:middle line:84% to be closed if the complement is open. 00:24:04.970 --> 00:24:07.670 align:middle line:90% 00:24:07.670 --> 00:24:12.030 align:middle line:84% So that's how you define a closed subset. 00:24:12.030 --> 00:24:22.140 align:middle line:84% And so, again, as we observed, since the empty set and x are 00:24:22.140 --> 00:24:31.980 align:middle line:84% both open, the complement of the empty set is the whole set. 00:24:31.980 --> 00:24:35.460 align:middle line:84% So it means that x and the complement of the whole set 00:24:35.460 --> 00:24:36.820 align:middle line:90% is the empty set. 00:24:36.820 --> 00:24:40.340 align:middle line:84% It means that those sets are both open-- 00:24:40.340 --> 00:24:43.490 align:middle line:90% are also both closed. 00:24:43.490 --> 00:24:55.660 align:middle line:90% 00:24:55.660 --> 00:25:01.900 align:middle line:84% So now there is another way-- you could have defined 00:25:01.900 --> 00:25:04.700 align:middle line:90% a closed subset another way. 00:25:04.700 --> 00:25:08.120 align:middle line:84% And this is because you have the following statement here. 00:25:08.120 --> 00:25:14.620 align:middle line:90% 00:25:14.620 --> 00:25:29.850 align:middle line:84% So a subset-- you have the following theorem that, again, 00:25:29.850 --> 00:25:32.130 align:middle line:90% you have a metric space. 00:25:32.130 --> 00:25:47.090 align:middle line:84% And C here, subset of x, is closed if and only if 00:25:47.090 --> 00:25:52.410 align:middle line:90% for all sequences in C-- 00:25:52.410 --> 00:26:12.450 align:middle line:84% if and only if for all sequences xn where 00:26:12.450 --> 00:26:20.810 align:middle line:84% xn is in C, if you have all sequences, all, let's say, 00:26:20.810 --> 00:26:22.310 align:middle line:90% convergent sequences-- 00:26:22.310 --> 00:26:28.250 align:middle line:90% 00:26:28.250 --> 00:26:36.730 align:middle line:84% so all convergent sequence where the sequence lie in C, 00:26:36.730 --> 00:26:55.190 align:middle line:90% then the limit x is also in C. 00:26:55.190 --> 00:27:00.770 align:middle line:84% So I claim that if you want to check that a set is closed, 00:27:00.770 --> 00:27:03.770 align:middle line:84% instead of checking that the complement is open, 00:27:03.770 --> 00:27:06.330 align:middle line:90% you could check that. 00:27:06.330 --> 00:27:09.370 align:middle line:84% And this seems like if you were literally 00:27:09.370 --> 00:27:14.010 align:middle line:84% going to check it on any given example, explicit example, 00:27:14.010 --> 00:27:16.040 align:middle line:90% you would never do it this way. 00:27:16.040 --> 00:27:18.540 align:middle line:84% But it's very useful for proving things. 00:27:18.540 --> 00:27:23.480 align:middle line:84% And so the thing is that instead of proving that a complement is 00:27:23.480 --> 00:27:25.720 align:middle line:84% open, instead you could just prove 00:27:25.720 --> 00:27:30.000 align:middle line:84% that whenever you have a sequence in that set, 00:27:30.000 --> 00:27:34.060 align:middle line:84% then is the limit also in that set? 00:27:34.060 --> 00:27:36.640 align:middle line:84% If that's the case for all sequences, 00:27:36.640 --> 00:27:38.840 align:middle line:90% then the set is closed. 00:27:38.840 --> 00:27:42.820 align:middle line:84% And so we need to-- so let's try to prove that. 00:27:42.820 --> 00:27:47.700 align:middle line:84% And again, it's useful theoretically. 00:27:47.700 --> 00:27:53.320 align:middle line:84% But in practice, this would be totally useless. 00:27:53.320 --> 00:27:55.330 align:middle line:84% But it is useful to proving things. 00:27:55.330 --> 00:27:57.920 align:middle line:90% 00:27:57.920 --> 00:28:03.050 align:middle line:84% And so there's two things we need to prove. 00:28:03.050 --> 00:28:07.480 align:middle line:90% 00:28:07.480 --> 00:28:09.000 align:middle line:90% So there's two directions. 00:28:09.000 --> 00:28:21.360 align:middle line:84% So assume first that C is closed. 00:28:21.360 --> 00:28:30.310 align:middle line:90% 00:28:30.310 --> 00:28:39.100 align:middle line:84% And let xn in C be a convergent sequence. 00:28:39.100 --> 00:28:46.310 align:middle line:90% 00:28:46.310 --> 00:29:19.420 align:middle line:84% We want to show that the limit x is also in C. 00:29:19.420 --> 00:29:43.220 align:middle line:84% Since xn converges to x, then for all r greater than 0, 00:29:43.220 --> 00:29:58.460 align:middle line:84% there exists xn in the ball here of radius r around 00:29:58.460 --> 00:30:00.920 align:middle line:90% x because it's converging. 00:30:00.920 --> 00:30:08.620 align:middle line:84% So this means that for each r, you must have this. 00:30:08.620 --> 00:30:10.590 align:middle line:90% But now you see that. 00:30:10.590 --> 00:30:29.570 align:middle line:84% And if x was not in C, if x is not in C, since c is closed, 00:30:29.570 --> 00:30:32.540 align:middle line:90% the complement is open. 00:30:32.540 --> 00:30:35.130 align:middle line:90% 00:30:35.130 --> 00:30:40.310 align:middle line:84% And so this means that now x is assumed to be in a complement. 00:30:40.310 --> 00:30:43.130 align:middle line:84% But that means that there exists some r 00:30:43.130 --> 00:30:49.290 align:middle line:84% so that the ball here of radius of that r 00:30:49.290 --> 00:30:52.310 align:middle line:84% would be contained in the complement. 00:30:52.310 --> 00:31:00.150 align:middle line:90% 00:31:00.150 --> 00:31:04.930 align:middle line:84% But this means that there is no xn in that ball 00:31:04.930 --> 00:31:09.440 align:middle line:84% because all of the xn's are in C. 00:31:09.440 --> 00:31:12.380 align:middle line:84% And this here is entirely contained in the complement. 00:31:12.380 --> 00:31:15.980 align:middle line:90% 00:31:15.980 --> 00:31:18.060 align:middle line:90% So this is the contradiction. 00:31:18.060 --> 00:31:23.480 align:middle line:84% So if it is closed and you have a convergent sequence, 00:31:23.480 --> 00:31:26.680 align:middle line:84% that sequence is lying in this closed set, 00:31:26.680 --> 00:31:33.340 align:middle line:84% then the limit must be also contained in the set 00:31:33.340 --> 00:31:35.520 align:middle line:84% because, otherwise, there would be-- 00:31:35.520 --> 00:31:38.000 align:middle line:84% the limit would lie outside, and there would be 00:31:38.000 --> 00:31:40.260 align:middle line:90% a little ball centered at that. 00:31:40.260 --> 00:31:44.840 align:middle line:84% So there was no element in the sequence that was in that ball. 00:31:44.840 --> 00:31:49.960 align:middle line:84% But then that sequence couldn't converge to the center. 00:31:49.960 --> 00:31:51.320 align:middle line:90% And the other thing-- 00:31:51.320 --> 00:31:52.980 align:middle line:90% so this is one direction. 00:31:52.980 --> 00:32:03.120 align:middle line:84% The other direction that we need to prove is that if-- 00:32:03.120 --> 00:32:05.360 align:middle line:90% again, it's useless. 00:32:05.360 --> 00:32:07.520 align:middle line:84% You would never go about to prove 00:32:07.520 --> 00:32:10.310 align:middle line:84% that a set is closed in an example, 00:32:10.310 --> 00:32:14.510 align:middle line:84% but it is useful more theoretically. 00:32:14.510 --> 00:32:17.630 align:middle line:84% So suppose you have a set, and you 00:32:17.630 --> 00:32:22.710 align:middle line:84% knew that whenever you have a sequence in that set, 00:32:22.710 --> 00:32:25.870 align:middle line:84% then the limit that is convergent-- 00:32:25.870 --> 00:32:30.050 align:middle line:84% then the limit is also in the set. 00:32:30.050 --> 00:32:38.818 align:middle line:84% So we also need to show the converse and, again, 00:32:38.818 --> 00:32:39.360 align:middle line:90% the converse. 00:32:39.360 --> 00:32:43.270 align:middle line:90% 00:32:43.270 --> 00:33:06.310 align:middle line:84% So we need to show that if for all sequences xn with line 00:33:06.310 --> 00:33:34.390 align:middle line:84% C n is convergent, then the limit x is also in C. Then 00:33:34.390 --> 00:33:41.650 align:middle line:90% C is in fact closed. 00:33:41.650 --> 00:33:43.090 align:middle line:90% So that would be the converse. 00:33:43.090 --> 00:33:47.570 align:middle line:84% So we prove if and only if it has this property. 00:33:47.570 --> 00:33:50.390 align:middle line:90% So now let's assume that-- 00:33:50.390 --> 00:33:57.070 align:middle line:84% so suppose we have this property that all convergent sequences 00:33:57.070 --> 00:34:02.550 align:middle line:84% that the limit convergent sequence in the set. 00:34:02.550 --> 00:34:05.500 align:middle line:84% Then the limit was also in the set. 00:34:05.500 --> 00:34:09.699 align:middle line:84% Now let's prove that the set is closed. 00:34:09.699 --> 00:34:12.139 align:middle line:84% So to prove the set is closed, we 00:34:12.139 --> 00:34:16.100 align:middle line:84% need to show that the complement is open. 00:34:16.100 --> 00:34:36.369 align:middle line:84% So we need to show that the complement here is open. 00:34:36.369 --> 00:34:41.500 align:middle line:90% 00:34:41.500 --> 00:34:48.360 align:middle line:84% So take x here in the complement. 00:34:48.360 --> 00:34:56.100 align:middle line:90% 00:34:56.100 --> 00:35:03.700 align:middle line:90% If for all r greater than 0-- 00:35:03.700 --> 00:35:06.810 align:middle line:90% 00:35:06.810 --> 00:35:10.470 align:middle line:90% so we need to prove that-- 00:35:10.470 --> 00:35:13.330 align:middle line:90% sorry, let me say it like this. 00:35:13.330 --> 00:35:16.210 align:middle line:84% So we take some x, and we need to show 00:35:16.210 --> 00:35:26.690 align:middle line:84% that there exists some r strictly bigger than 0 00:35:26.690 --> 00:35:33.010 align:middle line:84% so that this ball here is contained in the complement. 00:35:33.010 --> 00:35:34.630 align:middle line:90% That's what we need to prove. 00:35:34.630 --> 00:35:38.210 align:middle line:84% And we need to prove this for each x in the complement. 00:35:38.210 --> 00:35:52.970 align:middle line:84% Now suppose that if this was not the case, 00:35:52.970 --> 00:35:56.450 align:middle line:90% then I can look at for each n-- 00:35:56.450 --> 00:36:08.880 align:middle line:84% then for each natural number here, you can look at the ball. 00:36:08.880 --> 00:36:12.760 align:middle line:84% It's always going to be centered at this x. 00:36:12.760 --> 00:36:18.640 align:middle line:84% But I'm looking at it with radii that is smaller and smaller. 00:36:18.640 --> 00:36:24.040 align:middle line:84% If for each n, natural number, if that ball 00:36:24.040 --> 00:36:26.360 align:middle line:84% was not contained in the complement-- 00:36:26.360 --> 00:36:29.760 align:middle line:84% so this means that for each n, you could pick-- 00:36:29.760 --> 00:36:34.720 align:middle line:90% so pick xn in that set. 00:36:34.720 --> 00:36:38.460 align:middle line:90% So you just choose. 00:36:38.460 --> 00:36:43.380 align:middle line:84% Because we're assuming that the complement is not closed. 00:36:43.380 --> 00:36:44.840 align:middle line:90% It's not open. 00:36:44.840 --> 00:36:47.880 align:middle line:84% So it means that there would exist at x so 00:36:47.880 --> 00:36:52.500 align:middle line:84% that no ball here would be entirely in the complement. 00:36:52.500 --> 00:36:56.800 align:middle line:84% So this means that this one here would not be entirely contained 00:36:56.800 --> 00:36:57.700 align:middle line:90% in complement. 00:36:57.700 --> 00:37:00.830 align:middle line:84% So this means that there would be some xn. 00:37:00.830 --> 00:37:10.310 align:middle line:84% And this xn here would actually be in the set C. 00:37:10.310 --> 00:37:16.830 align:middle line:84% But now, of course, the distance here from xn 00:37:16.830 --> 00:37:22.910 align:middle line:84% to x, this distance here is strictly less than 1 over n. 00:37:22.910 --> 00:37:25.790 align:middle line:84% So this means that-- so this implies 00:37:25.790 --> 00:37:30.150 align:middle line:90% that the xn converges to x. 00:37:30.150 --> 00:37:32.350 align:middle line:84% And now the assumption was that whenever 00:37:32.350 --> 00:37:36.670 align:middle line:84% you have a sequence that was converging, these guys lie in C. 00:37:36.670 --> 00:37:39.350 align:middle line:84% Then the limit here also lie in C. 00:37:39.350 --> 00:37:44.430 align:middle line:84% So then you have, from the assumption that x here is in C. 00:37:44.430 --> 00:37:45.810 align:middle line:90% But this is the contradiction. 00:37:45.810 --> 00:37:51.870 align:middle line:84% You're assuming that x was not in C. 00:37:51.870 --> 00:37:58.220 align:middle line:84% And again, this here is really, really useful. 00:37:58.220 --> 00:38:02.560 align:middle line:84% But it's only useful of as a tool to proving things. 00:38:02.560 --> 00:38:06.380 align:middle line:84% It's not useful for checking whether an example is 00:38:06.380 --> 00:38:07.130 align:middle line:90% open or closed. 00:38:07.130 --> 00:38:15.220 align:middle line:90% 00:38:15.220 --> 00:38:19.740 align:middle line:84% Let me just quickly give you a couple of examples. 00:38:19.740 --> 00:38:21.985 align:middle line:84% I probably should have done that already that. 00:38:21.985 --> 00:38:25.380 align:middle line:90% 00:38:25.380 --> 00:38:28.140 align:middle line:90% If you're looking at-- 00:38:28.140 --> 00:38:30.700 align:middle line:90% so what are closed subsets? 00:38:30.700 --> 00:38:34.140 align:middle line:90% So closed subset could be-- 00:38:34.140 --> 00:38:36.660 align:middle line:90% so here's some examples-- 00:38:36.660 --> 00:38:37.530 align:middle line:90% closed subsets. 00:38:37.530 --> 00:38:42.540 align:middle line:90% 00:38:42.540 --> 00:38:49.420 align:middle line:84% So if it's an r, a closed subset is some interval, 00:38:49.420 --> 00:38:54.300 align:middle line:84% let's say, from 0 to 1, where both these points are-- 00:38:54.300 --> 00:38:55.960 align:middle line:90% both 0 and 1 is included. 00:38:55.960 --> 00:38:58.650 align:middle line:90% This has closed. 00:38:58.650 --> 00:38:59.950 align:middle line:90% And why is it closed? 00:38:59.950 --> 00:39:02.030 align:middle line:84% Because the complement here is open. 00:39:02.030 --> 00:39:04.350 align:middle line:84% It's easy to see that the complement is open. 00:39:04.350 --> 00:39:07.970 align:middle line:90% 00:39:07.970 --> 00:39:11.210 align:middle line:84% Something like if you include one of the endpoints 00:39:11.210 --> 00:39:14.720 align:middle line:84% but not the other, this is either closed or open. 00:39:14.720 --> 00:39:24.010 align:middle line:90% 00:39:24.010 --> 00:39:28.210 align:middle line:84% And also, if you're looking at the complement, 00:39:28.210 --> 00:39:31.210 align:middle line:84% say, from 0 of the complement of-- 00:39:31.210 --> 00:39:34.930 align:middle line:84% so if you're looking at this thing here, 00:39:34.930 --> 00:39:40.660 align:middle line:84% union, we'll say, from 2 to 3, this is also closed. 00:39:40.660 --> 00:39:49.970 align:middle line:90% 00:39:49.970 --> 00:39:57.290 align:middle line:84% And maybe one more example is if you just take a single point, 00:39:57.290 --> 00:39:58.240 align:middle line:90% this is closed. 00:39:58.240 --> 00:40:02.970 align:middle line:90% 00:40:02.970 --> 00:40:05.650 align:middle line:84% And again, you just check that the complement is open. 00:40:05.650 --> 00:40:08.730 align:middle line:90% 00:40:08.730 --> 00:40:14.570 align:middle line:84% So now the next thing, which I think 00:40:14.570 --> 00:40:16.730 align:middle line:84% we talked very briefly about that last time 00:40:16.730 --> 00:40:19.200 align:middle line:84% also, because we talked a little bit about closed subset. 00:40:19.200 --> 00:40:23.083 align:middle line:90% 00:40:23.083 --> 00:40:24.500 align:middle line:84% So you have the following theorem. 00:40:24.500 --> 00:40:30.090 align:middle line:90% 00:40:30.090 --> 00:40:33.650 align:middle line:84% So if you have a metric space here 00:40:33.650 --> 00:40:46.570 align:middle line:84% and you have a family C alpha of closed subset, 00:40:46.570 --> 00:40:52.410 align:middle line:84% then the intersection of these closed subset, 00:40:52.410 --> 00:40:54.470 align:middle line:90% this here, is also closed. 00:40:54.470 --> 00:41:00.800 align:middle line:90% 00:41:00.800 --> 00:41:08.360 align:middle line:84% And if you have just finitely many closed subset, 00:41:08.360 --> 00:41:16.750 align:middle line:84% then the union of finitely many closed subset is also closed. 00:41:16.750 --> 00:41:22.000 align:middle line:90% 00:41:22.000 --> 00:41:27.280 align:middle line:84% And so here, it's exactly the opposite thing as for open sets 00:41:27.280 --> 00:41:30.720 align:middle line:84% that you can take any intersection, 00:41:30.720 --> 00:41:33.400 align:middle line:90% and it's still closed. 00:41:33.400 --> 00:41:36.660 align:middle line:84% But if you take union, you can only do it for finitely many. 00:41:36.660 --> 00:41:38.440 align:middle line:90% So this is exactly the opposite. 00:41:38.440 --> 00:41:40.480 align:middle line:90% Why is it exactly the opposite? 00:41:40.480 --> 00:41:44.500 align:middle line:84% It's because in order to prove that something is closed, 00:41:44.500 --> 00:41:47.720 align:middle line:84% it means that the complement is open. 00:41:47.720 --> 00:41:55.870 align:middle line:84% And then we have these basic relations for-- 00:41:55.870 --> 00:41:57.600 align:middle line:90% so the proof of this again. 00:41:57.600 --> 00:42:02.830 align:middle line:90% 00:42:02.830 --> 00:42:05.390 align:middle line:84% And so these are these operations on set 00:42:05.390 --> 00:42:10.430 align:middle line:84% that we went through, that if you take x here 00:42:10.430 --> 00:42:16.670 align:middle line:84% and you take the complement of intersections of subsets. 00:42:16.670 --> 00:42:26.670 align:middle line:84% This thing here is the same as the union of the complement, 00:42:26.670 --> 00:42:27.830 align:middle line:90% like that. 00:42:27.830 --> 00:42:31.950 align:middle line:84% And you see that since this is closed, 00:42:31.950 --> 00:42:34.390 align:middle line:84% this means that the complement is open. 00:42:34.390 --> 00:42:37.810 align:middle line:84% And you can take any union of open sets, and it's still open. 00:42:37.810 --> 00:42:41.790 align:middle line:84% So that's why you could take any intersection. 00:42:41.790 --> 00:42:44.310 align:middle line:84% And so the other thing is that if you're 00:42:44.310 --> 00:42:50.220 align:middle line:84% looking to prove that this thing here is closed, 00:42:50.220 --> 00:42:54.320 align:middle line:84% then you're looking at the complement of this union here. 00:42:54.320 --> 00:42:59.020 align:middle line:90% 00:42:59.020 --> 00:43:03.280 align:middle line:84% But this thing here is now the intersection, 00:43:03.280 --> 00:43:16.060 align:middle line:84% the complement of the intersection of the C1 up to-- 00:43:16.060 --> 00:43:21.140 align:middle line:84% sorry, this is the complement of-- this 00:43:21.140 --> 00:43:29.880 align:middle line:84% is the intersection of these guys, like this. 00:43:29.880 --> 00:43:39.860 align:middle line:90% 00:43:39.860 --> 00:43:47.020 align:middle line:84% And you see here, since C1 up to Cn, since they are closed, 00:43:47.020 --> 00:43:48.770 align:middle line:90% these guys here are open. 00:43:48.770 --> 00:43:52.470 align:middle line:84% And you're looking at finite intersections of open sets, 00:43:52.470 --> 00:43:53.930 align:middle line:90% and that's open. 00:43:53.930 --> 00:43:57.530 align:middle line:84% So again, for closed sets, in terms 00:43:57.530 --> 00:43:59.570 align:middle line:84% of taking intersection and union, 00:43:59.570 --> 00:44:01.230 align:middle line:90% it's exactly the opposite. 00:44:01.230 --> 00:44:05.280 align:middle line:90% 00:44:05.280 --> 00:44:09.760 align:middle line:84% Now, the next thing I want to talk about is a cover. 00:44:09.760 --> 00:44:14.290 align:middle line:90% 00:44:14.290 --> 00:44:19.640 align:middle line:90% So have a metric space. 00:44:19.640 --> 00:44:25.130 align:middle line:90% 00:44:25.130 --> 00:44:27.190 align:middle line:90% So we have a metric space here. 00:44:27.190 --> 00:44:30.090 align:middle line:90% 00:44:30.090 --> 00:44:32.330 align:middle line:90% And we have a subset-- 00:44:32.330 --> 00:44:34.760 align:middle line:90% A here is a subset of x. 00:44:34.760 --> 00:44:37.690 align:middle line:90% 00:44:37.690 --> 00:44:40.050 align:middle line:90% And then we have a collection. 00:44:40.050 --> 00:44:55.040 align:middle line:84% And then you alpha is a family of subsets of x. 00:44:55.040 --> 00:45:15.600 align:middle line:84% And we say that u alpha is a cover of A 00:45:15.600 --> 00:45:24.720 align:middle line:84% if A is contained in the union of the u alphas. 00:45:24.720 --> 00:45:30.050 align:middle line:84% So this is what it means to cover that set by a family. 00:45:30.050 --> 00:45:35.600 align:middle line:90% 00:45:35.600 --> 00:45:50.480 align:middle line:84% A subcover of the cover u alpha is-- 00:45:50.480 --> 00:45:52.990 align:middle line:90% 00:45:52.990 --> 00:45:58.070 align:middle line:84% so imagine that you have a whole bunch of sets. 00:45:58.070 --> 00:46:02.670 align:middle line:84% And together, they cover A sub cover means 00:46:02.670 --> 00:46:06.750 align:middle line:84% that you don't need to take all of the u alphas, 00:46:06.750 --> 00:46:08.950 align:middle line:84% if you can just take some of them, 00:46:08.950 --> 00:46:12.370 align:middle line:90% and they will still cover it. 00:46:12.370 --> 00:46:28.190 align:middle line:84% So a cover is a subfamily of the given family 00:46:28.190 --> 00:46:45.030 align:middle line:84% u alpha such that the union of those still cover. 00:46:45.030 --> 00:46:52.100 align:middle line:90% 00:46:52.100 --> 00:46:57.070 align:middle line:84% So let's look at a couple of examples. 00:46:57.070 --> 00:47:02.620 align:middle line:90% 00:47:02.620 --> 00:47:05.460 align:middle line:84% I mean, if you take a family that 00:47:05.460 --> 00:47:10.440 align:middle line:84% covers something, then the family itself is a subcover. 00:47:10.440 --> 00:47:14.340 align:middle line:84% But that's a kind of trivial subcover. 00:47:14.340 --> 00:47:19.800 align:middle line:90% And so you could look at-- 00:47:19.800 --> 00:47:23.460 align:middle line:90% so just here's an example. 00:47:23.460 --> 00:47:28.180 align:middle line:84% Suppose that the metric space is x with the usual as r-- sorry, 00:47:28.180 --> 00:47:31.220 align:middle line:90% with the usual metric. 00:47:31.220 --> 00:47:35.680 align:middle line:84% And suppose now that the u alpha is actually Un's, 00:47:35.680 --> 00:47:40.580 align:middle line:84% and those are integral from minus n-- 00:47:40.580 --> 00:47:41.840 align:middle line:90% let me just make them open. 00:47:41.840 --> 00:47:42.620 align:middle line:90% It doesn't matter. 00:47:42.620 --> 00:47:45.740 align:middle line:84% For a cover, there's no assertion whether they're 00:47:45.740 --> 00:47:48.500 align:middle line:90% open or closed or neither. 00:47:48.500 --> 00:47:51.140 align:middle line:84% So if you're looking at those sets, 00:47:51.140 --> 00:47:55.900 align:middle line:84% well, if you take the union of all those sets, 00:47:55.900 --> 00:47:58.020 align:middle line:90% that gives you everything. 00:47:58.020 --> 00:48:03.220 align:middle line:84% So this family is a cover of the real line. 00:48:03.220 --> 00:48:08.680 align:middle line:84% Of course, if you take a proper subcover, meaning-- 00:48:08.680 --> 00:48:11.020 align:middle line:90% I mean, well, you could take a-- 00:48:11.020 --> 00:48:13.780 align:middle line:90% so this here cover all of it. 00:48:13.780 --> 00:48:16.220 align:middle line:84% But you see that this particular cover has 00:48:16.220 --> 00:48:21.060 align:middle line:90% the property that Un plus 1-- 00:48:21.060 --> 00:48:25.300 align:middle line:84% this is contained, the previous one. 00:48:25.300 --> 00:48:31.860 align:middle line:84% So you don't really need all of them to cover r. 00:48:31.860 --> 00:48:35.820 align:middle line:84% If you wanted to, you could start with some very large n, 00:48:35.820 --> 00:48:38.220 align:middle line:84% and then you could go on from there. 00:48:38.220 --> 00:48:43.890 align:middle line:84% But there wouldn't be finitely many of the Un that would cover. 00:48:43.890 --> 00:48:48.690 align:middle line:84% You need infinite money to cover because, otherwise, there 00:48:48.690 --> 00:48:53.740 align:middle line:84% would be some of last point and both interest. 00:48:53.740 --> 00:48:56.430 align:middle line:90% 00:48:56.430 --> 00:49:00.370 align:middle line:84% So that's a cover and a subcover. 00:49:00.370 --> 00:49:14.210 align:middle line:84% And an open cover is, then, where 00:49:14.210 --> 00:49:21.000 align:middle line:90% each u alpha is an open subset. 00:49:21.000 --> 00:49:29.570 align:middle line:90% 00:49:29.570 --> 00:49:33.070 align:middle line:84% I mean, you it's really distinct that-- 00:49:33.070 --> 00:49:40.680 align:middle line:90% 00:49:40.680 --> 00:49:42.480 align:middle line:90% so that's our open cover. 00:49:42.480 --> 00:49:48.480 align:middle line:84% So each of the sets that is in the covering has to be open. 00:49:48.480 --> 00:50:03.840 align:middle line:84% And the last definition we need is a finite subcover 00:50:03.840 --> 00:50:15.900 align:middle line:84% is a subcover consisting of finite sets. 00:50:15.900 --> 00:50:26.880 align:middle line:90% 00:50:26.880 --> 00:50:28.850 align:middle line:90% So that's a finite subcover. 00:50:28.850 --> 00:50:31.520 align:middle line:90% 00:50:31.520 --> 00:50:35.560 align:middle line:90% And so now this leads to the-- 00:50:35.560 --> 00:50:43.510 align:middle line:84% and again, in this example here, r where this here is a cover, 00:50:43.510 --> 00:50:45.050 align:middle line:90% there is no finite-- 00:50:45.050 --> 00:51:04.630 align:middle line:84% so note that this particular cover does not 00:51:04.630 --> 00:51:06.180 align:middle line:90% have a finite subcover. 00:51:06.180 --> 00:51:14.390 align:middle line:90% 00:51:14.390 --> 00:51:16.950 align:middle line:84% If you just take finitely many of these, 00:51:16.950 --> 00:51:19.910 align:middle line:84% they are not going to cover all of r. 00:51:19.910 --> 00:51:21.510 align:middle line:84% But actually, in this example, you 00:51:21.510 --> 00:51:25.210 align:middle line:84% see also that the cover is an open cover. 00:51:25.210 --> 00:51:28.130 align:middle line:90% 00:51:28.130 --> 00:51:34.950 align:middle line:84% So this now leads to the definition of compactness. 00:51:34.950 --> 00:51:39.770 align:middle line:90% So you have a metric space. 00:51:39.770 --> 00:51:47.540 align:middle line:90% 00:51:47.540 --> 00:51:52.580 align:middle line:90% So x comma d is a metric space. 00:51:52.580 --> 00:51:55.140 align:middle line:90% A here is a subset. 00:51:55.140 --> 00:52:44.170 align:middle line:84% And we say that A is compact if for every open cover of A 00:52:44.170 --> 00:52:46.000 align:middle line:90% there is a finite subcover. 00:52:46.000 --> 00:53:05.530 align:middle line:90% 00:53:05.530 --> 00:53:10.810 align:middle line:84% And the thing that is really useful is-- 00:53:10.810 --> 00:53:19.530 align:middle line:84% and we won't prove this today, but I'll prove it next time. 00:53:19.530 --> 00:53:30.840 align:middle line:84% So this is the following result. So why is this notion useful? 00:53:30.840 --> 00:53:35.080 align:middle line:90% 00:53:35.080 --> 00:53:38.360 align:middle line:84% So one of the key properties, when 00:53:38.360 --> 00:53:40.840 align:middle line:84% you're thinking about sequences, one 00:53:40.840 --> 00:53:46.320 align:middle line:84% of the key properties for sequences on the line 00:53:46.320 --> 00:53:50.900 align:middle line:84% was the Cauchy convergence theorem. 00:53:50.900 --> 00:53:52.420 align:middle line:84% And the Cauchy convergence theorem, 00:53:52.420 --> 00:53:54.780 align:middle line:84% again, said that whenever you have a Cauchy sequence, 00:53:54.780 --> 00:53:56.900 align:middle line:84% then it was automatically convergent, 00:53:56.900 --> 00:54:00.280 align:middle line:84% and it was often much easier to check that a sequence was 00:54:00.280 --> 00:54:03.880 align:middle line:84% a Cauchy sequence than to actually find the limit 00:54:03.880 --> 00:54:07.800 align:middle line:84% and prove that it's converging to that. 00:54:07.800 --> 00:54:10.300 align:middle line:84% But to prove the Cauchy convergence theorem, 00:54:10.300 --> 00:54:12.840 align:middle line:84% we needed the Bolzano-Weierstrass theorem. 00:54:12.840 --> 00:54:16.940 align:middle line:90% And so this is because-- 00:54:16.940 --> 00:54:20.240 align:middle line:84% so why is it useful, this notion of compactness? 00:54:20.240 --> 00:54:28.480 align:middle line:84% So it's useful because there's a metric version 00:54:28.480 --> 00:54:48.030 align:middle line:84% of the Bolzano-Weierstrass theorem for metric spaces. 00:54:48.030 --> 00:54:53.360 align:middle line:90% 00:54:53.360 --> 00:54:57.160 align:middle line:84% And so this is the following theorem. 00:54:57.160 --> 00:55:00.960 align:middle line:84% And again, I won't prove it now, but I'll prove it next time, 00:55:00.960 --> 00:55:09.640 align:middle line:84% that if you take a metric space, and A here is a subset 00:55:09.640 --> 00:55:30.552 align:middle line:84% and A is compact, then any sequence in A 00:55:30.552 --> 00:55:37.070 align:middle line:90% has a convergent subsequence. 00:55:37.070 --> 00:55:43.890 align:middle line:90% 00:55:43.890 --> 00:55:45.950 align:middle line:84% So remember that the Bolzano-Weierstrass theorem, 00:55:45.950 --> 00:55:54.870 align:middle line:84% again, said that if you take an upper bounded sequence in r, 00:55:54.870 --> 00:55:58.160 align:middle line:84% then there is a subsequence that converge. 00:55:58.160 --> 00:56:01.110 align:middle line:90% 00:56:01.110 --> 00:56:08.440 align:middle line:84% So now let's prove a few things about-- 00:56:08.440 --> 00:56:14.790 align:middle line:90% 00:56:14.790 --> 00:56:18.990 align:middle line:84% and let me also look at a couple of more examples. 00:56:18.990 --> 00:56:25.190 align:middle line:84% So the first thing I want to just discuss is that-- 00:56:25.190 --> 00:56:27.250 align:middle line:90% so this here is the key-- 00:56:27.250 --> 00:56:30.860 align:middle line:84% this is our key application of compactness 00:56:30.860 --> 00:56:32.580 align:middle line:84% that you have this metric version 00:56:32.580 --> 00:56:34.480 align:middle line:84% of the Bolzano-Weierstrass theorem. 00:56:34.480 --> 00:56:47.580 align:middle line:90% 00:56:47.580 --> 00:56:53.940 align:middle line:84% But let's prove a couple of things about compact subsets. 00:56:53.940 --> 00:56:56.760 align:middle line:90% So we have a metric space here. 00:56:56.760 --> 00:57:00.420 align:middle line:90% 00:57:00.420 --> 00:57:07.650 align:middle line:90% And I have A here is compact. 00:57:07.650 --> 00:57:12.340 align:middle line:90% 00:57:12.340 --> 00:57:17.780 align:middle line:84% Then I want to show that this implies two things. 00:57:17.780 --> 00:57:20.510 align:middle line:90% One is that A is bounded. 00:57:20.510 --> 00:57:28.810 align:middle line:90% 00:57:28.810 --> 00:57:31.780 align:middle line:84% And the second one is that A is closed. 00:57:31.780 --> 00:57:41.610 align:middle line:90% 00:57:41.610 --> 00:57:43.940 align:middle line:84% Let's prove first that A is bounded. 00:57:43.940 --> 00:58:00.430 align:middle line:90% 00:58:00.430 --> 00:58:04.690 align:middle line:84% So you could look at a very trivial cover of A. 00:58:04.690 --> 00:58:13.810 align:middle line:84% So you have here A. Well, obviously, 00:58:13.810 --> 00:58:20.530 align:middle line:84% if you take all balls, let's say, with radius 1, 00:58:20.530 --> 00:58:24.480 align:middle line:84% and you take all balls, but now the center 00:58:24.480 --> 00:58:30.920 align:middle line:90% here could be anything in A. 00:58:30.920 --> 00:58:36.400 align:middle line:84% For each element in A, you take a ball of radius 1 00:58:36.400 --> 00:58:38.720 align:middle line:90% around that point. 00:58:38.720 --> 00:58:42.360 align:middle line:84% But first of all, these balls are all open. 00:58:42.360 --> 00:58:46.340 align:middle line:84% And of course, this thing here, the union of these guys, 00:58:46.340 --> 00:58:49.140 align:middle line:84% well, the union contains, in particular, all the centers, 00:58:49.140 --> 00:58:55.400 align:middle line:84% but the centers is already A. So you trivially have this. 00:58:55.400 --> 00:59:06.720 align:middle line:84% Now since A is compact, this means 00:59:06.720 --> 00:59:11.100 align:middle line:84% that A must be that this is an open cover. 00:59:11.100 --> 00:59:16.840 align:middle line:84% So it means that finitely many of them already cover A, 00:59:16.840 --> 00:59:25.410 align:middle line:84% so you have finitely many, like this. 00:59:25.410 --> 00:59:30.750 align:middle line:84% But now I claim that this automatically 00:59:30.750 --> 00:59:35.470 align:middle line:84% implies that A is bounded because, you see, 00:59:35.470 --> 00:59:39.110 align:middle line:90% you have all these balls here. 00:59:39.110 --> 00:59:43.010 align:middle line:84% And A is somehow contained in the union of these guys. 00:59:43.010 --> 00:59:46.850 align:middle line:84% So this means that for any element-- 00:59:46.850 --> 00:59:49.350 align:middle line:90% so this means that for any x-- 00:59:49.350 --> 01:00:00.910 align:middle line:84% so if x here is in A, then there exists 01:00:00.910 --> 01:00:06.110 align:middle line:84% one of these finitely many i, where i is one of these finitely 01:00:06.110 --> 01:00:10.510 align:middle line:90% many so that the distance. 01:00:10.510 --> 01:00:16.030 align:middle line:90% to this xi is smaller than 1. 01:00:16.030 --> 01:00:20.230 align:middle line:84% We would like to say that a is contained in a large ball. 01:00:20.230 --> 01:00:36.820 align:middle line:90% 01:00:36.820 --> 01:00:42.580 align:middle line:84% Now you just pick one of these xi's. 01:00:42.580 --> 01:00:45.200 align:middle line:90% Let's pick the first one. 01:00:45.200 --> 01:00:52.050 align:middle line:84% And let's look at the maximum of these-- 01:00:52.050 --> 01:00:54.740 align:middle line:90% 01:00:54.740 --> 01:00:57.530 align:middle line:84% from the first one to the other one. 01:00:57.530 --> 01:01:02.580 align:middle line:90% 01:01:02.580 --> 01:01:05.820 align:middle line:90% And let's add 1 to that. 01:01:05.820 --> 01:01:10.160 align:middle line:84% You take the maximum of finitely many numbers. 01:01:10.160 --> 01:01:13.380 align:middle line:84% So this is a nice finitely many numbers. 01:01:13.380 --> 01:01:17.340 align:middle line:90% And you call this here r. 01:01:17.340 --> 01:01:24.290 align:middle line:84% Now you look at the ball of radius r around x1. 01:01:24.290 --> 01:01:26.710 align:middle line:90% And I want to show that any x-- 01:01:26.710 --> 01:01:29.270 align:middle line:90% 01:01:29.270 --> 01:01:44.650 align:middle line:84% so I want to show That for any x in A, 01:01:44.650 --> 01:01:50.790 align:middle line:84% then we have that x is in this ball, like that. 01:01:50.790 --> 01:02:15.510 align:middle line:90% 01:02:15.510 --> 01:02:21.490 align:middle line:84% So now we know that-- so since x here is, 01:02:21.490 --> 01:02:25.670 align:middle line:84% we know that x is contained in the union of this ball. 01:02:25.670 --> 01:02:36.650 align:middle line:84% So we know that x is contained in this ball here for some i, 01:02:36.650 --> 01:02:40.330 align:middle line:84% where it's one of these little n many i's. 01:02:40.330 --> 01:02:42.850 align:middle line:90% 01:02:42.850 --> 01:02:47.010 align:middle line:84% So this means now that the distance-- 01:02:47.010 --> 01:02:53.410 align:middle line:84% and so you know that the distance to this little i 01:02:53.410 --> 01:02:54.850 align:middle line:90% is at most 1. 01:02:54.850 --> 01:02:58.250 align:middle line:84% And so now if you're looking at the distance 01:02:58.250 --> 01:03:02.250 align:middle line:84% from x to the first one, then this here 01:03:02.250 --> 01:03:09.930 align:middle line:84% is less than the distance from this to xi 01:03:09.930 --> 01:03:14.810 align:middle line:84% plus the distance-- this is the triangle inequality again xi. 01:03:14.810 --> 01:03:16.640 align:middle line:84% Maybe let me just make very clear 01:03:16.640 --> 01:03:21.360 align:middle line:84% that this is the first one to the first one. 01:03:21.360 --> 01:03:24.160 align:middle line:90% This is the triangle inequality. 01:03:24.160 --> 01:03:28.360 align:middle line:90% But these distances here are-- 01:03:28.360 --> 01:03:41.920 align:middle line:90% 01:03:41.920 --> 01:03:44.600 align:middle line:84% this thing here is bounded by $1. 01:03:44.600 --> 01:03:49.120 align:middle line:84% It's actually strictly less than 1. 01:03:49.120 --> 01:03:53.420 align:middle line:84% So this is less than 1 plus this thing here. 01:03:53.420 --> 01:03:57.520 align:middle line:90% 01:03:57.520 --> 01:04:00.520 align:middle line:84% And you see, this thing here is, of course, this 01:04:00.520 --> 01:04:04.000 align:middle line:90% is how we define r. 01:04:04.000 --> 01:04:08.560 align:middle line:84% This is actually less than or equal to r. 01:04:08.560 --> 01:04:13.703 align:middle line:84% And so you have this thing here is strictly less than r. 01:04:13.703 --> 01:04:15.370 align:middle line:84% And that's what we want wanted to prove. 01:04:15.370 --> 01:04:18.810 align:middle line:90% 01:04:18.810 --> 01:04:22.950 align:middle line:84% So this proves that if you take, again, a metric space and you 01:04:22.950 --> 01:04:28.230 align:middle line:84% have a compact subset, then it's automatically bounded. 01:04:28.230 --> 01:04:29.870 align:middle line:90% What about the second one? 01:04:29.870 --> 01:04:33.430 align:middle line:90% Suppose you take a subset. 01:04:33.430 --> 01:04:35.990 align:middle line:90% Is it closed? 01:04:35.990 --> 01:04:38.780 align:middle line:84% Now let's use the theorem that we just proved. 01:04:38.780 --> 01:04:45.270 align:middle line:90% 01:04:45.270 --> 01:04:46.460 align:middle line:90% So we need to prove-- 01:04:46.460 --> 01:05:01.110 align:middle line:90% 01:05:01.110 --> 01:05:05.940 align:middle line:84% so let's prove that this A here is closed. 01:05:05.940 --> 01:05:23.460 align:middle line:90% 01:05:23.460 --> 01:05:27.200 align:middle line:90% So now if A is closed-- 01:05:27.200 --> 01:05:40.460 align:middle line:84% so to prove that A is closed by the theorem 01:05:40.460 --> 01:05:43.120 align:middle line:84% we just proved just a little while ago-- 01:05:43.120 --> 01:05:46.140 align:middle line:90% 01:05:46.140 --> 01:05:53.740 align:middle line:84% theorem we proved earlier, we need 01:05:53.740 --> 01:06:08.620 align:middle line:84% to show that if xn is a sequence in A and xn is converging 01:06:08.620 --> 01:06:36.930 align:middle line:84% to some x, we need to show that x is also in A. 01:06:36.930 --> 01:06:41.370 align:middle line:84% Suppose now we want to show that this leads to a contradiction. 01:06:41.370 --> 01:06:42.990 align:middle line:90% We want to get a contradiction. 01:06:42.990 --> 01:06:54.050 align:middle line:90% 01:06:54.050 --> 01:06:57.650 align:middle line:90% So you have this x here. 01:06:57.650 --> 01:07:02.390 align:middle line:84% And you're assuming that x is not in A. 01:07:02.390 --> 01:07:11.400 align:middle line:84% Yet you have a sequence in A that somehow tends to this-- 01:07:11.400 --> 01:07:14.670 align:middle line:84% this is the sequence xn, and it's heading to this element. 01:07:14.670 --> 01:07:19.520 align:middle line:90% 01:07:19.520 --> 01:07:31.540 align:middle line:84% Well, we saw earlier that we define something called E here, 01:07:31.540 --> 01:07:42.680 align:middle line:84% which is the x, the y in x, so that the distance from x to y 01:07:42.680 --> 01:07:45.080 align:middle line:90% is strictly bigger than r. 01:07:45.080 --> 01:07:48.940 align:middle line:84% Then we saw that this here is an open set. 01:07:48.940 --> 01:07:51.800 align:middle line:90% We show that this is open. 01:07:51.800 --> 01:07:55.320 align:middle line:90% This is for each r. 01:07:55.320 --> 01:08:02.880 align:middle line:84% So I'm going to now find a cover of A. I'm 01:08:02.880 --> 01:08:05.040 align:middle line:90% going to find an open cover. 01:08:05.040 --> 01:08:13.550 align:middle line:84% And the open cover is that for each integer, for each n here, 01:08:13.550 --> 01:08:18.710 align:middle line:90% I'm going to look at E 1 over n. 01:08:18.710 --> 01:08:23.710 align:middle line:84% So this means that I'm looking at the y 01:08:23.710 --> 01:08:30.430 align:middle line:84% so that the distance from y to this x here that I want to show 01:08:30.430 --> 01:08:37.370 align:middle line:84% is an A, that this thing here is bigger than 1 over n. 01:08:37.370 --> 01:08:41.950 align:middle line:90% 01:08:41.950 --> 01:08:48.750 align:middle line:84% But now I claim that A is contained 01:08:48.750 --> 01:08:59.630 align:middle line:84% in the union of these guys because, you see, 01:08:59.630 --> 01:09:06.830 align:middle line:84% actually, these guys here, if you have something that is not 01:09:06.830 --> 01:09:14.910 align:middle line:84% in any of these, this means that this inequality fails 01:09:14.910 --> 01:09:20.910 align:middle line:84% with each n, but this means that-- so if set 01:09:20.910 --> 01:09:30.310 align:middle line:84% is in the complement of the union of these guys, 01:09:30.310 --> 01:09:33.229 align:middle line:84% actually, then you have that this thing 01:09:33.229 --> 01:09:39.810 align:middle line:84% here is less than or equal to 1 over n for all n. 01:09:39.810 --> 01:09:43.149 align:middle line:90% 01:09:43.149 --> 01:09:44.430 align:middle line:90% You have that. 01:09:44.430 --> 01:09:47.750 align:middle line:84% That's a complement of the union. 01:09:47.750 --> 01:09:50.630 align:middle line:84% But if it's less than 1 over n for all n, 01:09:50.630 --> 01:09:51.860 align:middle line:90% it means that it must be 0. 01:09:51.860 --> 01:09:54.510 align:middle line:90% 01:09:54.510 --> 01:09:57.030 align:middle line:84% So this means that this distance here-- 01:09:57.030 --> 01:10:01.610 align:middle line:84% so you conclude that this distance is 0. 01:10:01.610 --> 01:10:07.580 align:middle line:84% But this means that this set here is actually your x. 01:10:07.580 --> 01:10:11.740 align:middle line:84% So the complement of these guys, the complement 01:10:11.740 --> 01:10:15.980 align:middle line:84% of this union here is simply just a point x. 01:10:15.980 --> 01:10:18.460 align:middle line:84% But you are assuming that x was not 01:10:18.460 --> 01:10:24.100 align:middle line:84% in your set A. The complement here, the union of these guys, 01:10:24.100 --> 01:10:29.740 align:middle line:84% we have proved now that the union covers everything but x. 01:10:29.740 --> 01:10:32.740 align:middle line:84% So in particular, it covers A because we're assuming 01:10:32.740 --> 01:10:38.700 align:middle line:90% that little x was not in A. 01:10:38.700 --> 01:10:44.460 align:middle line:84% So this is a nice open cover of A. 01:10:44.460 --> 01:10:46.980 align:middle line:84% But A was supposed to be compact. 01:10:46.980 --> 01:10:49.360 align:middle line:84% It means that finitely many of these cover. 01:10:49.360 --> 01:10:52.540 align:middle line:90% 01:10:52.540 --> 01:11:03.100 align:middle line:84% But now this family has this property that you see-- 01:11:03.100 --> 01:11:09.370 align:middle line:84% these guys here, if you're looking at E 1 over n, 01:11:09.370 --> 01:11:17.650 align:middle line:84% this thing here is contained in E 1 over n plus 1. 01:11:17.650 --> 01:11:21.170 align:middle line:84% If this number here is smaller, then it 01:11:21.170 --> 01:11:25.610 align:middle line:84% means that this set here is larger. 01:11:25.610 --> 01:11:27.330 align:middle line:90% So you have this-- 01:11:27.330 --> 01:11:30.890 align:middle line:84% and so you see that if finitely many-- 01:11:30.890 --> 01:11:36.690 align:middle line:84% if A here was contained in finitely many of these guys 01:11:36.690 --> 01:11:52.010 align:middle line:84% here, if it was contained in a union of finitely many, 01:11:52.010 --> 01:11:57.250 align:middle line:84% then you just take one of these n that is the largest. 01:11:57.250 --> 01:12:03.000 align:middle line:84% So this means that A here would be contained in some 1 01:12:03.000 --> 01:12:10.060 align:middle line:84% over for some i because there's one of these numbers-- 01:12:10.060 --> 01:12:13.400 align:middle line:84% these ni's, that would be the largest. 01:12:13.400 --> 01:12:19.640 align:middle line:90% But this means now that A-- 01:12:19.640 --> 01:12:28.640 align:middle line:84% that there is some lower bound for the distance to x in A. 01:12:28.640 --> 01:12:30.760 align:middle line:90% There's no elements that is-- 01:12:30.760 --> 01:12:35.640 align:middle line:84% so this means that there exists-- 01:12:35.640 --> 01:12:48.880 align:middle line:84% that does not exist, let me say, for all y in A. 01:12:48.880 --> 01:12:58.680 align:middle line:84% Then the distance here from y to x is strictly bigger than 1 01:12:58.680 --> 01:13:02.110 align:middle line:90% over this in i. 01:13:02.110 --> 01:13:04.870 align:middle line:90% This is just some fixed number. 01:13:04.870 --> 01:13:07.430 align:middle line:84% This is obviously po-- you know, right? 01:13:07.430 --> 01:13:11.870 align:middle line:84% But this means that you can't have a sequence in A that 01:13:11.870 --> 01:13:18.890 align:middle line:84% is converging to x because they all stay away from x. 01:13:18.890 --> 01:13:23.450 align:middle line:84% They stay a certain distance away from x. 01:13:23.450 --> 01:13:26.190 align:middle line:90% 01:13:26.190 --> 01:13:28.830 align:middle line:84% So this is the desired contradiction. 01:13:28.830 --> 01:13:49.510 align:middle line:84% Therefore, there is no sequence that converges to x and the xn 01:13:49.510 --> 01:13:50.470 align:middle line:90% on A. 01:13:50.470 --> 01:13:54.470 align:middle line:84% But that's what we assumed in the beginning. 01:13:54.470 --> 01:13:55.830 align:middle line:90% We wanted to check. 01:13:55.830 --> 01:14:00.980 align:middle line:84% Then whenever you have a sequence in An 01:14:00.980 --> 01:14:05.660 align:middle line:84% that is converging, then the limit was also. 01:14:05.660 --> 01:14:08.280 align:middle line:84% So that was supposed to have been such a sequence. 01:14:08.280 --> 01:14:11.220 align:middle line:84% So that's a desired contradiction. 01:14:11.220 --> 01:14:17.900 align:middle line:84% So this proved that if you take a compact metric space, then 01:14:17.900 --> 01:14:23.300 align:middle line:84% if you take a metric space and if you take a compact subset, 01:14:23.300 --> 01:14:26.380 align:middle line:84% the compact subset could, of course, be all of x. 01:14:26.380 --> 01:14:26.990 align:middle line:90% That's fine. 01:14:26.990 --> 01:14:30.620 align:middle line:90% 01:14:30.620 --> 01:14:33.380 align:middle line:84% But I want to allow more generally that just 01:14:33.380 --> 01:14:36.540 align:middle line:84% to say that just having a subset that is compact, 01:14:36.540 --> 01:14:39.780 align:middle line:84% this means that it's bounded and closed. 01:14:39.780 --> 01:14:41.510 align:middle line:90% If you are in Rn-- 01:14:41.510 --> 01:14:44.620 align:middle line:90% 01:14:44.620 --> 01:14:51.660 align:middle line:84% so you have a theorem, and we'll talk about that next time also, 01:14:51.660 --> 01:14:52.820 align:middle line:90% that this theorem-- 01:14:52.820 --> 01:14:55.820 align:middle line:84% let's say the converse is the case. 01:14:55.820 --> 01:14:59.050 align:middle line:90% But that is not actually-- 01:14:59.050 --> 01:15:02.130 align:middle line:84% that does not generalize to metric spaces. 01:15:02.130 --> 01:15:04.970 align:middle line:90% So it says theorem here. 01:15:04.970 --> 01:15:09.330 align:middle line:84% Well, I mean it does generalize, but not like this. 01:15:09.330 --> 01:15:13.890 align:middle line:84% This particular statement does not hold. 01:15:13.890 --> 01:15:23.050 align:middle line:84% So in Rn with the usual metric, say the usual Euclidean metric, 01:15:23.050 --> 01:15:23.550 align:middle line:90% if you will. 01:15:23.550 --> 01:15:29.970 align:middle line:84% You can just think about R. If A is 01:15:29.970 --> 01:15:42.020 align:middle line:84% compact is closed and bounded, then A is compact. 01:15:42.020 --> 01:15:46.010 align:middle line:90% 01:15:46.010 --> 01:15:50.950 align:middle line:84% And I wanted to talk about an example, 01:15:50.950 --> 01:15:54.010 align:middle line:84% but I don't think I have enough time 01:15:54.010 --> 01:15:59.220 align:middle line:90% to actually talk about that. 01:15:59.220 --> 01:16:03.525 align:middle line:84% Let's talk about this-- return to this next time 01:16:03.525 --> 01:16:05.150 align:middle line:84% because we only have four minutes left. 01:16:05.150 --> 01:16:10.840 align:middle line:90% 01:16:10.840 --> 01:16:19.794 align:middle line:84% And so, instead, let's prove a theorem here that-- 01:16:19.794 --> 01:16:22.640 align:middle line:90% 01:16:22.640 --> 01:16:28.960 align:middle line:84% if we take the theorem, if you have a metric space, 01:16:28.960 --> 01:16:35.840 align:middle line:84% A here is a subset of x is compact, 01:16:35.840 --> 01:16:43.860 align:middle line:84% suppose that C here is a subset of A and C here is [INAUDIBLE]. 01:16:43.860 --> 01:16:45.140 align:middle line:90% So this is, of course, in x. 01:16:45.140 --> 01:16:47.400 align:middle line:90% So C is also a subset of x. 01:16:47.400 --> 01:16:51.800 align:middle line:90% And C is closed. 01:16:51.800 --> 01:16:57.170 align:middle line:84% Then the claim is that then C is also compact. 01:16:57.170 --> 01:17:02.440 align:middle line:90% 01:17:02.440 --> 01:17:04.040 align:middle line:84% And we have three minutes, and we 01:17:04.040 --> 01:17:08.080 align:middle line:84% don't need three minutes because it's actually very quick. 01:17:08.080 --> 01:17:11.480 align:middle line:84% And so the thing is-- so proof of this-- 01:17:11.480 --> 01:17:15.280 align:middle line:84% so C here is assumed to be closed. 01:17:15.280 --> 01:17:22.200 align:middle line:84% So this means that since C is closed, 01:17:22.200 --> 01:17:26.450 align:middle line:84% this means that the complement of C is open. 01:17:26.450 --> 01:17:34.400 align:middle line:90% 01:17:34.400 --> 01:17:51.340 align:middle line:84% So now let's let O alpha be a cover of C. 01:17:51.340 --> 01:17:54.190 align:middle line:84% So O alpha is a cover of C, and now you 01:17:54.190 --> 01:17:58.430 align:middle line:84% can add the complement of C to it. 01:17:58.430 --> 01:18:02.990 align:middle line:90% So if you're looking at C-- 01:18:02.990 --> 01:18:11.070 align:middle line:84% so C here is now, of course, in fact, 01:18:11.070 --> 01:18:16.910 align:middle line:84% A here, you can think about A as-- so A is clearly 01:18:16.910 --> 01:18:28.110 align:middle line:84% contained in C union with the complement of C. 01:18:28.110 --> 01:18:34.650 align:middle line:84% And so this means that A here, the O alpha cover C-- 01:18:34.650 --> 01:18:39.350 align:middle line:84% so this is contained in the union of O alpha union 01:18:39.350 --> 01:18:41.510 align:middle line:90% with this last set. 01:18:41.510 --> 01:18:44.190 align:middle line:90% 01:18:44.190 --> 01:18:46.950 align:middle line:84% These guys are assumed to be open. 01:18:46.950 --> 01:18:49.950 align:middle line:84% This here is open because C is closed. 01:18:49.950 --> 01:18:54.220 align:middle line:84% So this means that you have now A is contained in-- 01:18:54.220 --> 01:18:58.260 align:middle line:84% this is an open cover of A. So this means that finitely many 01:18:58.260 --> 01:18:59.840 align:middle line:90% of these-- 01:18:59.840 --> 01:19:04.620 align:middle line:84% so this means that A is contained in like O1 union 01:19:04.620 --> 01:19:06.460 align:middle line:90% up to some On. 01:19:06.460 --> 01:19:11.900 align:middle line:84% And then, quite possibly, you would also need this one here. 01:19:11.900 --> 01:19:16.540 align:middle line:84% But now you see, but C here is contained in here. 01:19:16.540 --> 01:19:19.780 align:middle line:84% So this means that C is contained in this. 01:19:19.780 --> 01:19:22.460 align:middle line:84% This here was not part of the cover, 01:19:22.460 --> 01:19:26.340 align:middle line:84% but it's not needed if you just want to cover C because none 01:19:26.340 --> 01:19:30.780 align:middle line:84% of these elements cover anything in C. 01:19:30.780 --> 01:19:36.900 align:middle line:84% So this implies that C actually must be contained in O1 union 01:19:36.900 --> 01:19:42.020 align:middle line:84% up to On because this here does not cover anything in C. So 01:19:42.020 --> 01:19:44.100 align:middle line:90% this proves that C-- 01:19:44.100 --> 01:19:47.790 align:middle line:84% there was a finite subcover that covers C. 01:19:47.790 --> 01:19:50.000 align:middle line:90%