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Title: Open and Closed Sets
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Series: Functional Analysis
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YouTube-Title: Functional Analysis 3 | Open and Closed Sets
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Bright video: https://youtu.be/RYtE09eHeqI
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Dark video: https://youtu.be/he8rrQ1osk8
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Ad-free video: Watch Vimeo video
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Quiz: Test your knowledge
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Dark-PDF: Download PDF version of the dark video
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Print-PDF: Download printable PDF version
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Exercise Download PDF sheets
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Thumbnail (bright): Download PNG
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Thumbnail (dark): Download PNG
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Subtitle on GitHub: fa03_sub_eng.srt
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Timestamps
00:00 Introduction
00:37 Epsilon ball
01:35 Notions
06:24 Examples
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Subtitle in English
1 00:00:00,720 –> 00:00:02,166 Hello and welcome back to
2 00:00:02,198 –> 00:00:03,326 functional analysis.
3 00:00:03,438 –> 00:00:04,702 And as always, I want to
4 00:00:04,726 –> 00:00:05,966 thank all the nice people
5 00:00:06,038 –> 00:00:07,214 that support this channel
6 00:00:07,262 –> 00:00:08,810 on Steady or PayPal.
7 00:00:09,350 –> 00:00:10,446 We’ve reached part three
8 00:00:10,478 –> 00:00:11,678 in our course, and today
9 00:00:11,734 –> 00:00:13,238 we will talk about some important
10 00:00:13,334 –> 00:00:15,210 notions in metric spaces.
11 00:00:15,790 –> 00:00:17,342 Please recall a pair
12 00:00:17,406 –> 00:00:19,238 consisting of a set X and
13 00:00:19,254 –> 00:00:21,022 a metric d is called a
14 00:00:21,046 –> 00:00:21,930 metric space.
15 00:00:22,510 –> 00:00:24,054 We’ve already discussed that
16 00:00:24,102 –> 00:00:25,838 if you fix a point x in a
17 00:00:25,854 –> 00:00:27,374 metric space, you can look
18 00:00:27,422 –> 00:00:29,308 at all the other points that
19 00:00:29,324 –> 00:00:30,676 have the same distance from
20 00:00:30,708 –> 00:00:31,840 this point x.
21 00:00:32,300 –> 00:00:33,900 Seeing that in the common
22 00:00:33,940 –> 00:00:35,620 geometry of the plane, this
23 00:00:35,660 –> 00:00:36,908 would be a circle around
24 00:00:36,964 –> 00:00:37,560 x.
25 00:00:38,060 –> 00:00:39,212 Or in a three dimensional
26 00:00:39,276 –> 00:00:41,180 space it would be a sphere.
27 00:00:41,260 –> 00:00:42,720 Or you could call it a ball.
28 00:00:43,580 –> 00:00:44,892 Now exactly, this notion
29 00:00:44,916 –> 00:00:46,396 of a ball is what we want
30 00:00:46,428 –> 00:00:47,196 to generalize.
31 00:00:47,268 –> 00:00:48,960 For an abstract metric space,
32 00:00:49,620 –> 00:00:51,140 we write B epsilon
33 00:00:51,220 –> 00:00:53,172 x and call it the open
34 00:00:53,236 –> 00:00:54,484 epsilon ball around
35 00:00:54,572 –> 00:00:55,586 x.
36 00:00:55,768 –> 00:00:57,342 It is defined as all the
37 00:00:57,366 –> 00:00:59,214 points y in our metric
38 00:00:59,262 –> 00:01:01,126 space X that fulfill
39 00:01:01,238 –> 00:01:01,486 that
40 00:01:01,518 –> 00:01:03,310 The distance from x to
41 00:01:03,350 –> 00:01:05,222 y is less than
42 00:01:05,246 –> 00:01:07,010 a given radius epsilon.
43 00:01:07,430 –> 00:01:08,830 However, this means that
44 00:01:08,870 –> 00:01:10,582 in the picture it’s not the
45 00:01:10,606 –> 00:01:12,450 red line, it’s everything
46 00:01:12,830 –> 00:01:13,770 inside.
47 00:01:15,510 –> 00:01:16,838 Now please note that for
48 00:01:16,854 –> 00:01:18,422 a given positive radius
49 00:01:18,486 –> 00:01:20,430 epsilon and a fixed point
50 00:01:20,510 –> 00:01:22,028 x from the metric space,
51 00:01:22,174 –> 00:01:24,168 this epsilon ball is never
52 00:01:24,224 –> 00:01:26,112 empty, because at least the
53 00:01:26,136 –> 00:01:27,800 point x lies in this
54 00:01:27,840 –> 00:01:28,420 set.
55 00:01:29,200 –> 00:01:30,672 Using this definition of
56 00:01:30,696 –> 00:01:32,384 an epsilon ball, we can now
57 00:01:32,432 –> 00:01:34,096 talk about a lot of important
58 00:01:34,208 –> 00:01:36,140 notions in a metric space.
59 00:01:36,920 –> 00:01:38,336 The first one I want to show
60 00:01:38,368 –> 00:01:39,880 you is about open
61 00:01:39,960 –> 00:01:40,740 sets.
62 00:01:41,080 –> 00:01:42,424 You may already know what
63 00:01:42,472 –> 00:01:44,280 open means in Rn, but
64 00:01:44,320 –> 00:01:46,048 now we define it for arbitrary
65 00:01:46,104 –> 00:01:47,816 subsets of our metric
66 00:01:47,848 –> 00:01:49,270 space X.
67 00:01:49,640 –> 00:01:51,304 So let’s take this subset
68 00:01:51,352 –> 00:01:53,040 A and I will use this
69 00:01:53,080 –> 00:01:54,940 symbol to denote a subset.
70 00:01:55,760 –> 00:01:57,528 Now descriptively, openness
71 00:01:57,584 –> 00:01:59,360 should mean that if you
72 00:01:59,400 –> 00:02:01,208 are inside the set A, you
73 00:02:01,224 –> 00:02:02,704 should never see the boundary
74 00:02:02,752 –> 00:02:03,660 of the set.
75 00:02:04,240 –> 00:02:05,640 In other words, if you fix
76 00:02:05,680 –> 00:02:07,656 an arbitrary point x of the
77 00:02:07,688 –> 00:02:09,432 set A, there should be enough
78 00:02:09,496 –> 00:02:10,752 points in all directions
79 00:02:10,816 –> 00:02:12,592 around this point that also
80 00:02:12,656 –> 00:02:14,180 belong to the set A.
81 00:02:14,770 –> 00:02:16,426 Of course, in order to describe
82 00:02:16,458 –> 00:02:18,434 this we can use such an epsilon
83 00:02:18,482 –> 00:02:19,070 ball.
84 00:02:19,530 –> 00:02:21,202 We just have to choose positive
85 00:02:21,266 –> 00:02:22,826 epsilon, but we can choose
86 00:02:22,858 –> 00:02:24,750 it as small as we need it.
87 00:02:25,130 –> 00:02:26,690 And if we can do that for
88 00:02:26,730 –> 00:02:28,498 each point separately, then
89 00:02:28,554 –> 00:02:30,230 we have an open set A.
90 00:02:30,730 –> 00:02:32,002 Therefore, the definition
91 00:02:32,066 –> 00:02:33,682 reads like set is called
92 00:02:33,746 –> 00:02:35,522 open if for each point
93 00:02:35,586 –> 00:02:37,090 we can find such an epsilon
94 00:02:37,130 –> 00:02:37,710 ball.
95 00:02:38,370 –> 00:02:40,098 Of course, for each x you
96 00:02:40,114 –> 00:02:41,498 can choose another epsilon
97 00:02:41,554 –> 00:02:42,470 if needed.
98 00:02:43,100 –> 00:02:44,132 In the picture this would
99 00:02:44,156 –> 00:02:45,668 mean if you get closer to
100 00:02:45,684 –> 00:02:47,676 the boundary you need a smaller
101 00:02:47,708 –> 00:02:48,840 epsilon, of course.
102 00:02:49,500 –> 00:02:50,964 However, as long as you find
103 00:02:51,012 –> 00:02:52,924 for each point such an epsilon,
104 00:02:53,052 –> 00:02:54,600 we call the set open.
105 00:02:55,300 –> 00:02:56,700 Don’t worry, we have
106 00:02:56,740 –> 00:02:57,740 consistency here.
107 00:02:57,820 –> 00:02:59,700 The so called open balls
108 00:02:59,780 –> 00:03:01,420 are also open with this
109 00:03:01,460 –> 00:03:02,280 definition.
110 00:03:03,260 –> 00:03:04,652 This is a simple exercise
111 00:03:04,716 –> 00:03:06,160 you can do for yourself.
112 00:03:06,700 –> 00:03:07,692 Now, with your knowledge
113 00:03:07,716 –> 00:03:09,396 of open sets, you might also
114 00:03:09,468 –> 00:03:11,330 want to know what close sets
115 00:03:11,370 –> 00:03:11,950 are.
116 00:03:12,490 –> 00:03:13,882 However, before we do that,
117 00:03:13,946 –> 00:03:15,442 let’s talk about so called
118 00:03:15,506 –> 00:03:16,590 boundary points.
119 00:03:17,250 –> 00:03:18,730 Let’s take our arbitrary
120 00:03:18,770 –> 00:03:20,642 subset A again, and now
121 00:03:20,666 –> 00:03:22,130 we look in the picture maybe
122 00:03:22,170 –> 00:03:23,706 at points around
123 00:03:23,778 –> 00:03:24,390 here.
124 00:03:25,370 –> 00:03:26,874 At the moment it’s not important
125 00:03:26,962 –> 00:03:28,546 if the point we have chosen
126 00:03:28,618 –> 00:03:29,834 is an element of our set
127 00:03:29,882 –> 00:03:30,762 a or not.
128 00:03:30,866 –> 00:03:32,522 Of course, it’s a point in
129 00:03:32,546 –> 00:03:34,470 our whole metric space X.
130 00:03:35,410 –> 00:03:36,626 The important thing is that
131 00:03:36,658 –> 00:03:38,474 with these points we describe
132 00:03:38,522 –> 00:03:40,090 something that we could call
133 00:03:40,130 –> 00:03:41,550 the boundary of a.
134 00:03:42,030 –> 00:03:43,526 And of course we use the
135 00:03:43,558 –> 00:03:45,006 epsilon balls again for
136 00:03:45,038 –> 00:03:45,610 this.
137 00:03:46,390 –> 00:03:47,790 Now what you should see is
138 00:03:47,870 –> 00:03:49,230 if we have an epsilon ball
139 00:03:49,270 –> 00:03:50,926 around this point, then we
140 00:03:50,958 –> 00:03:52,918 hit points that are in a
141 00:03:53,054 –> 00:03:54,142 and we hit other points that
142 00:03:54,166 –> 00:03:54,930 are not.
143 00:03:55,510 –> 00:03:57,070 Of course, this clearly can
144 00:03:57,110 –> 00:03:57,690 happen.
145 00:03:58,190 –> 00:03:59,702 However, if it happens, no
146 00:03:59,726 –> 00:04:01,046 matter how small we choose
147 00:04:01,078 –> 00:04:02,606 the ball, then we are
148 00:04:02,638 –> 00:04:04,414 clearly on a thing we could
149 00:04:04,462 –> 00:04:05,530 call the boundary.
150 00:04:05,950 –> 00:04:07,366 Then the definition reads
151 00:04:07,438 –> 00:04:09,390 a point x from the whole
152 00:04:09,430 –> 00:04:10,142 metric space.
153 00:04:10,206 –> 00:04:11,884 X is called a boundary
154 00:04:11,932 –> 00:04:13,812 point for a if
155 00:04:13,876 –> 00:04:15,348 all open balls around
156 00:04:15,444 –> 00:04:17,132 x contain points
157 00:04:17,196 –> 00:04:18,836 from A and a complement of
158 00:04:18,868 –> 00:04:19,440 A.
159 00:04:20,140 –> 00:04:21,644 For the formula here we just
160 00:04:21,692 –> 00:04:23,132 use the intersection and
161 00:04:23,156 –> 00:04:24,860 say this one can’t be
162 00:04:24,900 –> 00:04:26,820 empty and also not the
163 00:04:26,860 –> 00:04:28,380 one when you use the complement
164 00:04:28,420 –> 00:04:29,240 of a.
165 00:04:29,580 –> 00:04:30,492 There are some important
166 00:04:30,556 –> 00:04:31,556 things I should point out
167 00:04:31,588 –> 00:04:31,948 here.
168 00:04:32,044 –> 00:04:33,748 First, a boundary point
169 00:04:33,844 –> 00:04:35,716 can be inside a set A or
170 00:04:35,748 –> 00:04:37,210 outside, and
171 00:04:37,250 –> 00:04:38,858 secondly, the notion boundary
172 00:04:38,914 –> 00:04:40,730 point makes only sense with
173 00:04:40,770 –> 00:04:42,314 respect to a given subset
174 00:04:42,362 –> 00:04:42,950 A.
175 00:04:43,330 –> 00:04:44,850 There is a symbol to denote
176 00:04:44,890 –> 00:04:46,442 all the boundary points which
177 00:04:46,466 –> 00:04:47,790 is used very often.
178 00:04:48,450 –> 00:04:50,418 Its this curved del in
179 00:04:50,434 –> 00:04:51,390 front of A.
180 00:04:51,970 –> 00:04:53,306 So we put all the points
181 00:04:53,378 –> 00:04:54,890 x that are boundary points
182 00:04:54,970 –> 00:04:56,906 for A into this set.
183 00:04:57,098 –> 00:04:59,066 Now you can remember an open
184 00:04:59,138 –> 00:05:00,778 set is exactly such a set
185 00:05:00,834 –> 00:05:02,058 where all the boundary points
186 00:05:02,114 –> 00:05:03,710 are outside of A.
187 00:05:04,310 –> 00:05:05,918 When you see this, then you
188 00:05:05,934 –> 00:05:07,262 immediately understand what
189 00:05:07,286 –> 00:05:08,850 a closed set should be.
190 00:05:09,470 –> 00:05:10,998 It should be a set where
191 00:05:11,054 –> 00:05:12,806 all the boundary points belong
192 00:05:12,878 –> 00:05:13,930 to this set.
193 00:05:14,590 –> 00:05:15,854 Using the same formula.
194 00:05:15,902 –> 00:05:17,782 This reads A is closed
195 00:05:17,886 –> 00:05:19,574 if and only if A with the
196 00:05:19,622 –> 00:05:21,430 union of the boundary is
197 00:05:21,470 –> 00:05:21,646 A
198 00:05:21,678 –> 00:05:22,250 again.
199 00:05:22,790 –> 00:05:23,966 However, that’s not what
200 00:05:23,998 –> 00:05:25,598 one uses as the definition
201 00:05:25,654 –> 00:05:26,678 for closed sets.
202 00:05:26,774 –> 00:05:28,530 The definition is much simpler.
203 00:05:29,240 –> 00:05:31,120 A subset A in X is now
204 00:05:31,160 –> 00:05:32,984 called closed if the
205 00:05:33,032 –> 00:05:34,832 complement in X, which
206 00:05:34,856 –> 00:05:36,560 is AC, is
207 00:05:36,600 –> 00:05:37,180 open.
208 00:05:37,800 –> 00:05:39,072 This makes sense because
209 00:05:39,136 –> 00:05:40,680 the boundary points of A
210 00:05:40,720 –> 00:05:42,136 and a complement are exactly
211 00:05:42,168 –> 00:05:42,940 the same.
212 00:05:43,400 –> 00:05:44,696 And this just means that
213 00:05:44,728 –> 00:05:45,960 all the boundary points belong
214 00:05:46,000 –> 00:05:47,540 to A and not AC.
215 00:05:48,320 –> 00:05:49,992 Now the last notion for today
216 00:05:50,096 –> 00:05:51,320 will be the so called
217 00:05:51,400 –> 00:05:52,300 closure.
218 00:05:52,680 –> 00:05:54,368 The name already tells you
219 00:05:54,464 –> 00:05:56,448 if you start with an arbitrary
220 00:05:56,504 –> 00:05:58,098 subset A, what you want to
221 00:05:58,114 –> 00:05:59,466 get out is a closed
222 00:05:59,498 –> 00:06:00,310 subset.
223 00:06:00,850 –> 00:06:01,874 How to get this?
224 00:06:01,922 –> 00:06:03,666 You might already know, you
225 00:06:03,698 –> 00:06:05,066 just add all the missing
226 00:06:05,098 –> 00:06:06,650 boundary points, so you form
227 00:06:06,690 –> 00:06:08,670 the union with the boundary.
228 00:06:09,210 –> 00:06:10,410 And this is what we call
229 00:06:10,450 –> 00:06:12,306 the closure of A, and
230 00:06:12,338 –> 00:06:14,066 we denote that with A
231 00:06:14,178 –> 00:06:15,110 overline.
232 00:06:16,210 –> 00:06:17,714 Now please remember, this
233 00:06:17,762 –> 00:06:19,490 always defines a closed set.
234 00:06:19,610 –> 00:06:21,130 Indeed, it’s the smallest
235 00:06:21,170 –> 00:06:23,050 closed set that still contains
236 00:06:23,130 –> 00:06:25,022 A. Okay, now
237 00:06:25,046 –> 00:06:26,126 I would suggest that the
238 00:06:26,158 –> 00:06:27,822 closure of this video is
239 00:06:27,846 –> 00:06:28,890 an example.
240 00:06:29,430 –> 00:06:31,006 It shouldn’t be too complicated.
241 00:06:31,078 –> 00:06:32,414 So let’s choose a metric
242 00:06:32,462 –> 00:06:33,998 space consisting of real
243 00:06:34,054 –> 00:06:34,810 numbers.
244 00:06:35,390 –> 00:06:36,966 X is now defined as all the
245 00:06:36,998 –> 00:06:38,550 numbers between one and three,
246 00:06:38,630 –> 00:06:40,430 where three is included and
247 00:06:40,470 –> 00:06:42,170 all numbers larger than four.
248 00:06:42,710 –> 00:06:44,158 And the metric is just defined
249 00:06:44,214 –> 00:06:45,942 as the normal distance function
250 00:06:46,006 –> 00:06:47,730 we have for real numbers.
251 00:06:48,390 –> 00:06:49,926 Okay, let’s start considering
252 00:06:49,958 –> 00:06:51,158 some subsets of
253 00:06:51,254 –> 00:06:53,242 X, and the first
254 00:06:53,306 –> 00:06:54,658 one is the interval from
255 00:06:54,714 –> 00:06:56,226 one to three, which is of
256 00:06:56,258 –> 00:06:57,706 course a nice subset of
257 00:06:57,738 –> 00:06:58,310 X.
258 00:06:58,930 –> 00:07:00,626 My question for you is now,
259 00:07:00,738 –> 00:07:02,218 is this also an open
260 00:07:02,274 –> 00:07:02,870 set?
261 00:07:04,370 –> 00:07:05,570 So this is how you should
262 00:07:05,610 –> 00:07:06,586 visualize the set.
263 00:07:06,658 –> 00:07:08,594 And now we look at each point
264 00:07:08,682 –> 00:07:10,642 here and try to find an
265 00:07:10,666 –> 00:07:12,290 epsilon ball around this
266 00:07:12,330 –> 00:07:12,910 point.
267 00:07:13,530 –> 00:07:15,074 Now you see, this is
268 00:07:15,122 –> 00:07:16,914 possible for all x
269 00:07:17,042 –> 00:07:18,810 that are in A, but not
270 00:07:18,850 –> 00:07:19,430 three.
271 00:07:19,870 –> 00:07:21,830 What you can do is just look
272 00:07:21,870 –> 00:07:23,246 at the distance, what should
273 00:07:23,278 –> 00:07:24,518 be the boundary here, left
274 00:07:24,574 –> 00:07:26,206 and right, and choose the
275 00:07:26,238 –> 00:07:27,462 minimal you have.
276 00:07:27,646 –> 00:07:29,102 And if you want, you can
277 00:07:29,166 –> 00:07:30,630 make that even smaller.
278 00:07:30,710 –> 00:07:32,206 To get to the picture here,
279 00:07:32,358 –> 00:07:34,170 maybe you divide by two,
280 00:07:35,310 –> 00:07:36,998 and then the epsilon ball
281 00:07:37,094 –> 00:07:38,646 around x is
282 00:07:38,678 –> 00:07:40,062 indeed exactly
283 00:07:40,166 –> 00:07:42,050 inside the set a.
284 00:07:42,790 –> 00:07:44,102 However, if we want to have
285 00:07:44,126 –> 00:07:45,946 an open set, we need this
286 00:07:45,978 –> 00:07:47,042 property also for the
287 00:07:47,066 –> 00:07:48,030 point 3
288 00:07:48,650 –> 00:07:50,322 ok, so let’s write down an
289 00:07:50,346 –> 00:07:51,946 epsilon ball, maybe with
290 00:07:51,978 –> 00:07:53,190 radius one.
291 00:07:54,290 –> 00:07:55,322 These are all the points
292 00:07:55,386 –> 00:07:57,226 y and x, where the distance
293 00:07:57,338 –> 00:07:59,150 is less than one.
294 00:08:00,050 –> 00:08:01,354 So let’s write down that
295 00:08:01,402 –> 00:08:02,282 as an interval.
296 00:08:02,386 –> 00:08:03,922 We already know that the
297 00:08:03,946 –> 00:08:05,634 interval two to three
298 00:08:05,762 –> 00:08:07,114 fulfills this property
299 00:08:07,162 –> 00:08:07,802 here.
300 00:08:07,986 –> 00:08:09,234 And the question is now,
301 00:08:09,322 –> 00:08:10,714 are there any other points
302 00:08:10,802 –> 00:08:12,154 that have distance less than
303 00:08:12,202 –> 00:08:13,550 one from three?
304 00:08:13,970 –> 00:08:15,442 It can’t be anything here
305 00:08:15,466 –> 00:08:16,962 in this area because the
306 00:08:16,986 –> 00:08:18,330 points are too far away from
307 00:08:18,370 –> 00:08:19,026 three.
308 00:08:19,218 –> 00:08:20,586 And also in this interval
309 00:08:20,618 –> 00:08:22,110 we don’t find any points
310 00:08:22,650 –> 00:08:23,954 because the distance from
311 00:08:24,002 –> 00:08:25,514 three to four is already
312 00:08:25,602 –> 00:08:27,106 one, but four is not
313 00:08:27,138 –> 00:08:28,642 included in
314 00:08:28,666 –> 00:08:29,162 summary.
315 00:08:29,266 –> 00:08:30,762 Indeed, these are all the
316 00:08:30,786 –> 00:08:32,230 points we find in X.
317 00:08:32,610 –> 00:08:34,506 However, this is a subset
318 00:08:34,538 –> 00:08:35,310 of a.
319 00:08:35,850 –> 00:08:37,242 So our conclusion is
320 00:08:37,266 –> 00:08:39,098 indeed A is an open
321 00:08:39,154 –> 00:08:39,750 set.
322 00:08:41,750 –> 00:08:43,350 Okay, so this is an important
323 00:08:43,430 –> 00:08:44,650 thing to get today.
324 00:08:45,230 –> 00:08:46,958 The question openness makes
325 00:08:47,014 –> 00:08:48,582 only sense if you know what
326 00:08:48,606 –> 00:08:50,070 the surrounding universe,
327 00:08:50,110 –> 00:08:51,862 the whole metric space X
328 00:08:51,926 –> 00:08:52,530 is.
329 00:08:53,110 –> 00:08:54,550 Otherwise you won’t be able
330 00:08:54,590 –> 00:08:56,294 to calculate the epsilon ball
331 00:08:56,422 –> 00:08:57,730 in X itself.
332 00:08:58,350 –> 00:09:00,022 In this case, the epsilon ball
333 00:09:00,126 –> 00:09:01,530 around three
334 00:09:02,030 –> 00:09:03,406 has only one side.
335 00:09:03,558 –> 00:09:04,890 So it looks like this.
336 00:09:05,760 –> 00:09:07,160 Now, what you can also show
337 00:09:07,240 –> 00:09:08,912 is that the set A here
338 00:09:09,016 –> 00:09:10,620 is also a closed set.
339 00:09:11,080 –> 00:09:12,408 This is what you also should
340 00:09:12,464 –> 00:09:13,340 immediately remember:
341 00:09:14,200 –> 00:09:16,104 Openness and closedness are
342 00:09:16,152 –> 00:09:17,540 not opposites.
343 00:09:18,200 –> 00:09:19,480 Surely you can have both
344 00:09:19,520 –> 00:09:20,376 at the same time.
345 00:09:20,488 –> 00:09:21,528 But it can also happen that
346 00:09:21,544 –> 00:09:23,280 a set is neither closed nor
347 00:09:23,320 –> 00:09:23,900 open.
348 00:09:24,400 –> 00:09:25,840 Okay, so let’s do a last
349 00:09:25,880 –> 00:09:26,576 example here.
350 00:09:26,608 –> 00:09:28,256 So this is our set C, one
351 00:09:28,288 –> 00:09:29,952 to two, where two is included.
352 00:09:30,056 –> 00:09:31,376 And I want to calculate the
353 00:09:31,408 –> 00:09:32,920 boundary of C.
354 00:09:33,330 –> 00:09:34,818 So this is our drawing for
355 00:09:34,834 –> 00:09:35,850 the set C.
356 00:09:36,010 –> 00:09:37,170 And you can immediately see
357 00:09:37,210 –> 00:09:38,634 that for all points below
358 00:09:38,722 –> 00:09:40,154 two, you can do the same
359 00:09:40,202 –> 00:09:41,670 thing as before here,
360 00:09:42,490 –> 00:09:43,970 which means you get an epsilon
361 00:09:44,010 –> 00:09:45,778 ball which is completely
362 00:09:45,834 –> 00:09:47,354 inside the set C itself.
363 00:09:47,482 –> 00:09:49,350 So it’s not a boundary point.
364 00:09:50,130 –> 00:09:51,442 Which means the only point
365 00:09:51,506 –> 00:09:52,898 we have to consider now is
366 00:09:52,914 –> 00:09:54,194 the .2 itself.
367 00:09:54,362 –> 00:09:55,890 And then you see, immediately,
368 00:09:55,970 –> 00:09:57,474 if we look at an epsilon ball
369 00:09:57,522 –> 00:09:59,144 around two, we will hit
370 00:09:59,192 –> 00:10:00,840 points here on the left and
371 00:10:00,880 –> 00:10:02,060 also on the right.
372 00:10:03,040 –> 00:10:04,752 Now there are points on the
373 00:10:04,776 –> 00:10:05,848 right that’s different from
374 00:10:05,864 –> 00:10:06,820 the three before.
375 00:10:08,000 –> 00:10:09,744 Hence our boundary is just
376 00:10:09,792 –> 00:10:11,384 the, point 2 nothing
377 00:10:11,432 –> 00:10:12,020 more.
378 00:10:12,560 –> 00:10:13,720 And to conclude the whole
379 00:10:13,760 –> 00:10:15,176 video, maybe we also write
380 00:10:15,208 –> 00:10:16,520 down the closure of C.
381 00:10:16,640 –> 00:10:18,620 And you see, it’s C itself,
382 00:10:19,040 –> 00:10:20,744 which means the set is
383 00:10:20,792 –> 00:10:21,660 closed.
384 00:10:22,200 –> 00:10:23,992 Okay, now I hope you understand
385 00:10:24,096 –> 00:10:25,528 these notions now a little
386 00:10:25,584 –> 00:10:26,180 better.
387 00:10:26,750 –> 00:10:28,046 And in the next video I will
388 00:10:28,078 –> 00:10:29,750 explain how we deal with
389 00:10:29,790 –> 00:10:31,770 them when we use sequences.
390 00:10:32,710 –> 00:10:34,022 So, thanks for listening
391 00:10:34,086 –> 00:10:35,406 and see you next time.
392 00:10:35,518 –> 00:10:35,790 Bye.
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Quiz Content
Q1: What is the correct definition for the open $\varepsilon$-ball $B_\varepsilon(x)$ in a metric space $(X,d)$?
A1: ${ y \in X \mid d(x,y) \neq 0 }$
A2: ${ y \in X \mid d(x,y) > \varepsilon }$
A3: ${ y \in X \mid d(x,y) < \varepsilon }$
A4: ${ y \in X \mid d(x,y) \leq \varepsilon }$
Q2: Let $(X,d)$ be a metric space. Is there an open $\varepsilon$-ball $B_\varepsilon(x)$ with $x \in X$ and $\varepsilon > 0$ which is empty?
A1: No!
A2: Yes!
Q3: Let $X = (1,5]$ and $d(x,y) = |x-y|$. Which of the following sets is open?
A1: $(1,5]$
A2: $(1,4]$
A3: $[2,4)$
A4: $[2,4]$
Q4: Let $X = (1,5]$ and $d(x,y) = |x-y|$. Which of the following sets is not closed?
A1: $(1,5]$
A2: $(1,4]$
A3: $[2,4)$
A4: $[2,4]$
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Last update: 2024-10
