-
Title: Interior, Exterior, Boundary, Closure
-
Series: Manifolds
-
YouTube-Title: Manifolds 2 | Interior, Exterior, Boundary, Closure
-
Bright video: https://youtu.be/nMcQvQ-vBEA
-
Dark video: https://youtu.be/l3YKEXNJzJU
-
Ad-free video: Watch Vimeo video
-
Quiz: Test your knowledge
-
Dark-PDF: Download PDF version of the dark video
-
Print-PDF: Download printable PDF version
-
Thumbnail (bright): Download PNG
-
Thumbnail (dark): Download PNG
-
Subtitle on GitHub: mf02_sub_eng.srt
-
Other languages: German version
-
Timestamps (n/a)
-
Subtitle in English
1 00:00:00,850 –> 00:00:02,630 Hello and welcome back to
2 00:00:02,640 –> 00:00:03,779 manifolds.
3 00:00:04,420 –> 00:00:05,730 And before we start, I want
4 00:00:05,739 –> 00:00:07,079 to thank all the nice people
5 00:00:07,090 –> 00:00:08,119 who support this channel
6 00:00:08,130 –> 00:00:09,829 and Steady via paypal or by
7 00:00:09,840 –> 00:00:10,479 other means.
8 00:00:11,220 –> 00:00:12,560 Now, in today’s part two,
9 00:00:12,569 –> 00:00:14,100 we continue talking about
10 00:00:14,109 –> 00:00:15,640 the topic of topology,
11 00:00:16,379 –> 00:00:18,129 namely, we introduce notions
12 00:00:18,139 –> 00:00:19,870 like interior, closure
13 00:00:19,879 –> 00:00:21,110 and related stuff
14 00:00:21,879 –> 00:00:22,379 for this
15 00:00:22,389 –> 00:00:23,809 Please recall that the
16 00:00:23,819 –> 00:00:25,750 topology on a set X is a
17 00:00:25,760 –> 00:00:27,700 collection of subsets that
18 00:00:27,709 –> 00:00:29,600 satisfies three rules.
19 00:00:30,030 –> 00:00:31,979 First, the smallest and the
20 00:00:31,989 –> 00:00:33,700 largest subset have to be
21 00:00:33,709 –> 00:00:34,360 involved.
22 00:00:35,020 –> 00:00:36,900 Secondly, you cannot leave
23 00:00:36,909 –> 00:00:38,860 the topology by finitely many
24 00:00:38,869 –> 00:00:39,919 intersections.
25 00:00:40,419 –> 00:00:42,349 And thirdly, you also cannot
26 00:00:42,360 –> 00:00:44,279 leave it with any unions.
27 00:00:45,279 –> 00:00:47,020 So these are the three properties
28 00:00:47,029 –> 00:00:48,979 of a topology T and you
29 00:00:48,990 –> 00:00:50,599 also should remember that
30 00:00:50,610 –> 00:00:52,439 the elements of T are called
31 00:00:52,450 –> 00:00:53,380 open sets.
32 00:00:54,229 –> 00:00:56,169 Moreover, the set X
33 00:00:56,180 –> 00:00:58,169 together with a chosen topology
34 00:00:58,180 –> 00:00:59,610 is called a topological
35 00:00:59,619 –> 00:01:00,240 space.
36 00:01:01,029 –> 00:01:02,700 Hence, it’s a space where
37 00:01:02,709 –> 00:01:04,540 the notion of open sets makes
38 00:01:04,550 –> 00:01:05,050 sense.
39 00:01:05,819 –> 00:01:07,699 Of course, we will also discuss
40 00:01:07,709 –> 00:01:09,180 examples of topological
41 00:01:09,190 –> 00:01:10,150 spaces here.
42 00:01:10,889 –> 00:01:12,660 However, before we do this,
43 00:01:12,669 –> 00:01:14,019 we first need more
44 00:01:14,029 –> 00:01:14,860 definitions.
45 00:01:15,680 –> 00:01:17,300 Indeed, in a topological
46 00:01:17,309 –> 00:01:19,220 space points in X
47 00:01:19,230 –> 00:01:20,739 can have special names.
48 00:01:21,610 –> 00:01:22,970 Therefore, here let’s fix
49 00:01:22,980 –> 00:01:24,139 a topological space
50 00:01:24,150 –> 00:01:25,529 (X,T) and a
51 00:01:25,540 –> 00:01:27,389 subset S in X
52 00:01:28,260 –> 00:01:30,000 here, please note S could
53 00:01:30,010 –> 00:01:31,750 be an element of the topology
54 00:01:31,760 –> 00:01:33,260 but it does not have to be
55 00:01:34,010 –> 00:01:34,510 OK.
56 00:01:34,519 –> 00:01:35,910 We want to talk about names
57 00:01:35,919 –> 00:01:36,709 for points.
58 00:01:36,720 –> 00:01:38,559 So let’s fix the point P
59 00:01:38,569 –> 00:01:39,430 in X.
60 00:01:40,330 –> 00:01:41,830 Now all the names we give
61 00:01:41,839 –> 00:01:43,790 for P are to be read with
62 00:01:43,800 –> 00:01:45,339 respect to the fix set
63 00:01:45,349 –> 00:01:45,989 S.
64 00:01:46,790 –> 00:01:48,430 Therefore, our visualization
65 00:01:48,440 –> 00:01:49,529 should look like this.
66 00:01:49,540 –> 00:01:51,269 We have the whole space X
67 00:01:51,819 –> 00:01:53,330 and inside we have the
68 00:01:53,339 –> 00:01:54,519 subset S
69 00:01:55,269 –> 00:01:56,889 then one possibility for
70 00:01:56,900 –> 00:01:58,489 P is that P is an
71 00:01:58,500 –> 00:01:59,830 element of S.
72 00:02:00,879 –> 00:02:02,339 Of course, this alone is
73 00:02:02,349 –> 00:02:04,050 not so interesting, but we
74 00:02:04,059 –> 00:02:05,269 could have even more.
75 00:02:06,069 –> 00:02:08,050 And then we talk of an interior
76 00:02:08,059 –> 00:02:09,169 point of S.
77 00:02:10,089 –> 00:02:11,830 So what this exactly means
78 00:02:11,839 –> 00:02:13,309 we can now define
79 00:02:14,179 –> 00:02:15,690 indeed what we need is an
80 00:02:15,699 –> 00:02:17,570 open set and maybe let’s
81 00:02:17,580 –> 00:02:18,770 call it U here.
82 00:02:19,660 –> 00:02:21,119 Now this open set should
83 00:02:21,130 –> 00:02:22,910 contain the point P and it
84 00:02:22,919 –> 00:02:24,639 should also lie completely
85 00:02:24,649 –> 00:02:26,110 inside the set S
86 00:02:27,050 –> 00:02:28,440 and if we have these two
87 00:02:28,449 –> 00:02:30,339 properties, we call P
88 00:02:30,350 –> 00:02:32,059 an interior point of S.
89 00:02:33,110 –> 00:02:34,750 So you see the crucial thing
90 00:02:34,759 –> 00:02:36,750 here is that we find a suitable
91 00:02:36,759 –> 00:02:38,490 open set from the topology
92 00:02:38,500 –> 00:02:40,119 T OK.
93 00:02:40,130 –> 00:02:41,279 Now, it might not surprise
94 00:02:41,289 –> 00:02:42,990 you when we have interior
95 00:02:43,000 –> 00:02:44,759 points of S, we also
96 00:02:44,770 –> 00:02:46,660 have exterior points of S.
97 00:02:47,470 –> 00:02:48,649 Of course, the name
98 00:02:48,800 –> 00:02:50,369 suggests in our picture
99 00:02:50,380 –> 00:02:52,080 here, the point P should
100 00:02:52,089 –> 00:02:53,690 lie outside of S.
101 00:02:54,479 –> 00:02:56,449 However, as before we
102 00:02:56,460 –> 00:02:57,919 want to lie outside,
103 00:02:57,929 –> 00:02:59,860 even with an open set U.
104 00:03:00,589 –> 00:03:02,289 So visually speaking, we
105 00:03:02,300 –> 00:03:03,809 want some distance in both
106 00:03:03,820 –> 00:03:05,500 cases from the boundary of
107 00:03:05,509 –> 00:03:05,929 S.
108 00:03:06,720 –> 00:03:08,300 Hence, we need an open set
109 00:03:08,309 –> 00:03:09,899 U such that there’s no
110 00:03:09,910 –> 00:03:11,300 overlap with S.
111 00:03:12,369 –> 00:03:14,320 Therefore, this exactly means
112 00:03:14,330 –> 00:03:16,110 that P is an interior
113 00:03:16,119 –> 00:03:17,679 point of the complement of
114 00:03:17,690 –> 00:03:19,630 S or with the same
115 00:03:19,639 –> 00:03:21,389 formula as before we find
116 00:03:21,399 –> 00:03:23,149 an open set U such that
117 00:03:23,160 –> 00:03:24,940 P is an element of U
118 00:03:24,949 –> 00:03:26,830 and U is a subset of X
119 00:03:26,839 –> 00:03:27,830 without S
120 00:03:28,800 –> 00:03:29,300 OK.
121 00:03:29,309 –> 00:03:30,679 Now we have two important
122 00:03:30,690 –> 00:03:31,419 terms here.
123 00:03:31,589 –> 00:03:33,020 And you see in our picture,
124 00:03:33,100 –> 00:03:34,339 we have described all the
125 00:03:34,350 –> 00:03:36,240 points inside the set S
126 00:03:36,380 –> 00:03:38,300 and all outside of S.
127 00:03:39,279 –> 00:03:41,100 Therefore, the points missing
128 00:03:41,110 –> 00:03:42,860 are the ones on the boundary
129 00:03:42,869 –> 00:03:43,509 of S.
130 00:03:44,080 –> 00:03:45,490 For this reason, we could
131 00:03:45,500 –> 00:03:47,369 define P to be a boundary
132 00:03:47,380 –> 00:03:48,899 point of S if it’s
133 00:03:48,910 –> 00:03:50,660 neither an interior point
134 00:03:50,669 –> 00:03:52,380 nor an exterior point.
135 00:03:53,289 –> 00:03:54,660 However, of course, we can
136 00:03:54,669 –> 00:03:56,300 also immediately describe
137 00:03:56,309 –> 00:03:57,960 this with opens set
138 00:03:58,080 –> 00:03:59,750 U. It
139 00:03:59,759 –> 00:04:01,199 simply means that no matter
140 00:04:01,210 –> 00:04:03,050 which opens set you, we choose,
141 00:04:03,169 –> 00:04:04,690 we always have an overlap
142 00:04:04,699 –> 00:04:06,550 with S and the complement
143 00:04:06,559 –> 00:04:07,110 of S.
144 00:04:07,979 –> 00:04:09,779 Hence, we can write for all
145 00:04:09,789 –> 00:04:11,320 open sets U with the
146 00:04:11,330 –> 00:04:13,080 property that P is an element
147 00:04:13,089 –> 00:04:13,820 of U.
148 00:04:13,889 –> 00:04:15,779 We have U
149 00:04:15,789 –> 00:04:17,190 intersected with S is non-
150 00:04:17,279 –> 00:04:18,750 empty and U
151 00:04:18,760 –> 00:04:20,519 intersected with X without
152 00:04:20,529 –> 00:04:22,399 S is also non-empty.
153 00:04:23,269 –> 00:04:24,980 Then such a point P with
154 00:04:24,989 –> 00:04:26,720 this property, we call a
155 00:04:26,730 –> 00:04:28,390 boundary point of S.
156 00:04:29,309 –> 00:04:29,660 OK.
157 00:04:29,670 –> 00:04:30,690 Now, you might think that
158 00:04:30,700 –> 00:04:32,339 we have all the names, but
159 00:04:32,350 –> 00:04:33,540 I still want to include a
160 00:04:33,549 –> 00:04:34,410 last one.
161 00:04:35,250 –> 00:04:36,679 Indeed, this one is often
162 00:04:36,690 –> 00:04:37,799 important when we want to
163 00:04:37,809 –> 00:04:39,320 deal with limits and it’s
164 00:04:39,329 –> 00:04:41,000 called accumulation point
165 00:04:41,010 –> 00:04:42,269 of the set S.
166 00:04:42,929 –> 00:04:44,390 And it simply means that
167 00:04:44,399 –> 00:04:46,079 the point P is not
168 00:04:46,089 –> 00:04:47,630 isolated from the rest of
169 00:04:47,640 –> 00:04:48,500 the set S.
170 00:04:49,320 –> 00:04:51,220 Hence, as before, we can
171 00:04:51,230 –> 00:04:52,850 describe this, when we look
172 00:04:52,859 –> 00:04:54,739 at all open sets U that
173 00:04:54,750 –> 00:04:56,140 contain the point P
174 00:04:57,019 –> 00:04:58,799 and then we want that something
175 00:04:58,809 –> 00:05:00,480 from the set S remains.
176 00:05:01,140 –> 00:05:03,049 Or in other words, the intersection
177 00:05:03,059 –> 00:05:04,570 with the set S should not
178 00:05:04,579 –> 00:05:05,750 be the empty set
179 00:05:06,709 –> 00:05:07,140 Here,
180 00:05:07,149 –> 00:05:08,540 the crucial thing is that
181 00:05:08,549 –> 00:05:10,140 this works no matter which
182 00:05:10,149 –> 00:05:12,029 open set U around P
183 00:05:12,040 –> 00:05:12,660 we choose.
184 00:05:13,549 –> 00:05:14,910 For this reason, you immediately
185 00:05:14,920 –> 00:05:16,890 see an exterior point
186 00:05:16,899 –> 00:05:18,489 can never be an accumulation
187 00:05:18,500 –> 00:05:18,929 point.
188 00:05:19,940 –> 00:05:20,440 OK.
189 00:05:20,450 –> 00:05:22,160 So here we have four important
190 00:05:22,170 –> 00:05:24,019 names for points that are
191 00:05:24,029 –> 00:05:25,410 defined with respect to a
192 00:05:25,420 –> 00:05:26,700 chosen subset
193 00:05:26,709 –> 00:05:27,000 S.
194 00:05:27,890 –> 00:05:29,750 Therefore, I would say please
195 00:05:29,760 –> 00:05:30,589 remember them.
196 00:05:31,579 –> 00:05:32,820 However, now we’re having
197 00:05:32,829 –> 00:05:34,589 these names for points, we
198 00:05:34,600 –> 00:05:36,239 are also able to define
199 00:05:36,250 –> 00:05:37,480 names for sets.
200 00:05:38,269 –> 00:05:39,959 But don’t worry, these are
201 00:05:39,970 –> 00:05:41,450 not complicated anymore.
202 00:05:42,339 –> 00:05:43,760 For example, for the set
203 00:05:43,769 –> 00:05:45,450 S, we can define
204 00:05:45,459 –> 00:05:46,559 S circle.
205 00:05:47,299 –> 00:05:48,910 This one is simply the collection
206 00:05:48,920 –> 00:05:50,790 of all points P in X
207 00:05:50,799 –> 00:05:52,730 that fulfill that P is an
208 00:05:52,739 –> 00:05:54,339 interior point of S.
209 00:05:55,279 –> 00:05:56,980 Therefore, S circle is
210 00:05:56,989 –> 00:05:58,730 called the interior of
211 00:05:58,739 –> 00:05:59,250 S.
212 00:06:00,339 –> 00:06:01,959 So the interior is defined
213 00:06:01,970 –> 00:06:03,540 for a subset and it gives
214 00:06:03,549 –> 00:06:04,940 us a new subset.
215 00:06:05,929 –> 00:06:07,309 And now it might not surprise
216 00:06:07,320 –> 00:06:08,869 you that we can do a similar
217 00:06:08,880 –> 00:06:10,459 thing for all the other points
218 00:06:10,470 –> 00:06:10,769 here.
219 00:06:11,750 –> 00:06:13,630 Hence, the next thing will
220 00:06:13,640 –> 00:06:15,359 be the exterior of S
221 00:06:16,250 –> 00:06:17,519 However, there, we don’t
222 00:06:17,529 –> 00:06:18,720 have a special symbol.
223 00:06:18,730 –> 00:06:20,429 We just write Ext of
224 00:06:20,440 –> 00:06:22,250 S, then we
225 00:06:22,260 –> 00:06:23,529 collect all the exterior
226 00:06:23,540 –> 00:06:25,290 points of S and call this
227 00:06:25,299 –> 00:06:26,929 set the exterior of
228 00:06:26,940 –> 00:06:28,790 S. OK.
229 00:06:28,890 –> 00:06:30,549 The next important subset
230 00:06:30,559 –> 00:06:32,429 will be the boundary of S.
231 00:06:33,260 –> 00:06:35,010 Indeed, this one is denoted
232 00:06:35,019 –> 00:06:36,570 with a curved lowercase
233 00:06:36,579 –> 00:06:37,019 d.
234 00:06:37,850 –> 00:06:39,459 It’s the same symbol we would
235 00:06:39,470 –> 00:06:40,820 also use for partial
236 00:06:40,829 –> 00:06:41,670 derivatives.
237 00:06:42,369 –> 00:06:44,220 However, here, dS denotes
238 00:06:44,230 –> 00:06:45,959 a whole set namely the
239 00:06:45,970 –> 00:06:47,269 one with all the boundary
240 00:06:47,279 –> 00:06:48,190 points of S.
241 00:06:48,970 –> 00:06:50,649 And not so surprising, this
242 00:06:50,660 –> 00:06:52,040 one is called the boundary
243 00:06:52,049 –> 00:06:52,709 of S.
244 00:06:53,480 –> 00:06:54,859 Hence, only one set
245 00:06:54,869 –> 00:06:56,619 remains the one about the
246 00:06:56,630 –> 00:06:57,880 accumulation points.
247 00:06:58,579 –> 00:06:59,850 And for this one, we have
248 00:06:59,859 –> 00:07:01,359 a rather strange notation,
249 00:07:01,369 –> 00:07:02,920 we call it S prime.
250 00:07:03,820 –> 00:07:05,250 Therefore, for a set, the
251 00:07:05,260 –> 00:07:06,940 line in the upper index means
252 00:07:06,950 –> 00:07:08,500 that we have all the accumulation
253 00:07:08,510 –> 00:07:09,660 points in the set.
254 00:07:10,579 –> 00:07:12,049 In fact, one often calls
255 00:07:12,059 –> 00:07:13,720 this the derivative of the
256 00:07:13,730 –> 00:07:14,570 set S.
257 00:07:15,309 –> 00:07:16,640 Another term we will use
258 00:07:16,649 –> 00:07:17,970 here is that this is the
259 00:07:17,980 –> 00:07:19,649 derived set of S.
260 00:07:20,670 –> 00:07:21,959 Now I would say what you
261 00:07:21,970 –> 00:07:23,489 should see here is that we
262 00:07:23,500 –> 00:07:24,859 have a lot of labels, you
263 00:07:24,869 –> 00:07:26,130 should know when doing
264 00:07:26,140 –> 00:07:26,959 topology.
265 00:07:27,619 –> 00:07:28,809 Actually, there are even
266 00:07:28,820 –> 00:07:29,970 more than just the four I
267 00:07:29,980 –> 00:07:30,880 showed you here.
268 00:07:31,029 –> 00:07:32,630 And the last one I really
269 00:07:32,640 –> 00:07:33,309 need to show you.
270 00:07:33,320 –> 00:07:34,859 Now, it’s an
271 00:07:34,869 –> 00:07:35,769 important one.
272 00:07:35,820 –> 00:07:37,290 It’s called the closure of
273 00:07:37,299 –> 00:07:37,799 S.
274 00:07:38,359 –> 00:07:39,660 And you see it has a nice
275 00:07:39,670 –> 00:07:41,630 notation, we just overline
276 00:07:41,640 –> 00:07:42,179 the set.
277 00:07:43,019 –> 00:07:44,179 Now, the definition is not
278 00:07:44,190 –> 00:07:45,880 so complicated, we just take
279 00:07:45,890 –> 00:07:47,839 the original set S and then
280 00:07:47,850 –> 00:07:49,279 take the union with the
281 00:07:49,290 –> 00:07:50,970 boundary and
282 00:07:50,980 –> 00:07:52,820 then the set is what we call
283 00:07:52,829 –> 00:07:54,209 the closure of S.
284 00:07:55,000 –> 00:07:56,850 So I already told you this
285 00:07:56,859 –> 00:07:58,179 is the last definition I
286 00:07:58,190 –> 00:07:59,500 want to show you today
287 00:08:00,320 –> 00:08:01,119 at this point.
288 00:08:01,130 –> 00:08:02,679 I really think it’s helpful
289 00:08:02,690 –> 00:08:04,079 to look at an example.
290 00:08:04,910 –> 00:08:06,279 So let’s take an example
291 00:08:06,290 –> 00:08:08,070 of a topology which is not
292 00:08:08,079 –> 00:08:08,649 so common.
293 00:08:09,420 –> 00:08:11,160 However, the set X can be
294 00:08:11,170 –> 00:08:11,959 very common.
295 00:08:12,000 –> 00:08:13,519 So let’s take the real number
296 00:08:13,529 –> 00:08:13,920 line.
297 00:08:14,890 –> 00:08:16,459 On the other hand, T should
298 00:08:16,470 –> 00:08:18,230 not be the standard topology.
299 00:08:18,329 –> 00:08:19,570 So let’s define it in the
300 00:08:19,579 –> 00:08:20,369 following way.
301 00:08:21,359 –> 00:08:23,170 First, we already know the
302 00:08:23,179 –> 00:08:24,890 empty set and the real number
303 00:08:24,899 –> 00:08:26,329 line should be included in
304 00:08:26,339 –> 00:08:27,130 the topology.
305 00:08:28,029 –> 00:08:29,660 However, all the other
306 00:08:29,670 –> 00:08:31,410 open sets should be half-
307 00:08:31,420 –> 00:08:32,700 bounded intervals.
308 00:08:33,570 –> 00:08:35,400 Therefore, each non-trivial
309 00:08:35,409 –> 00:08:36,900 open set is such an
310 00:08:36,909 –> 00:08:38,229 interval that starts with
311 00:08:38,239 –> 00:08:39,840 a real number a and goes
312 00:08:39,849 –> 00:08:40,580 to infinity
313 00:08:41,630 –> 00:08:41,940 here.
314 00:08:41,950 –> 00:08:43,419 Please note it’s important
315 00:08:43,429 –> 00:08:44,830 that the left boundary a
316 00:08:44,840 –> 00:08:46,780 here is not included in the
317 00:08:46,789 –> 00:08:47,380 interval.
318 00:08:48,190 –> 00:08:49,530 Now, it’s a good exercise
319 00:08:49,539 –> 00:08:51,260 for you to check that all
320 00:08:51,270 –> 00:08:53,039 the three rules of a topology
321 00:08:53,049 –> 00:08:54,049 are fulfilled here.
322 00:08:54,979 –> 00:08:56,400 Then with this knowledge,
323 00:08:56,409 –> 00:08:58,349 we can take a subset S and
324 00:08:58,359 –> 00:08:59,900 look at the sets we defined
325 00:08:59,909 –> 00:09:00,440 above.
326 00:09:01,309 –> 00:09:03,119 In fact, the subset si have
327 00:09:03,130 –> 00:09:04,869 in mind is the interval
328 00:09:04,880 –> 00:09:06,150 0 to 1.
329 00:09:07,489 –> 00:09:09,159 Also here, the zero and
330 00:09:09,169 –> 00:09:10,989 one are not included in the
331 00:09:11,000 –> 00:09:11,390 set.
332 00:09:12,159 –> 00:09:13,760 Now, the first thing we can
333 00:09:13,770 –> 00:09:15,539 note here is that this set
334 00:09:15,549 –> 00:09:16,700 is not open.
335 00:09:17,710 –> 00:09:19,409 So maybe you find it strange
336 00:09:19,419 –> 00:09:20,289 to say this.
337 00:09:20,369 –> 00:09:22,010 But you immediately see this
338 00:09:22,020 –> 00:09:23,900 set is not included in our
339 00:09:23,909 –> 00:09:25,450 collection of open sets.
340 00:09:26,469 –> 00:09:28,349 It’s simply not such an interval
341 00:09:28,359 –> 00:09:28,609 here.
342 00:09:29,609 –> 00:09:30,820 Therefore, the next thing
343 00:09:30,830 –> 00:09:32,530 we can see here is that S
344 00:09:32,539 –> 00:09:34,309 has no interior points at
345 00:09:34,320 –> 00:09:34,659 all.
346 00:09:35,460 –> 00:09:36,719 This is simply because we
347 00:09:36,729 –> 00:09:38,280 don’t find any non-
348 00:09:38,390 –> 00:09:40,150 empty open subset that
349 00:09:40,159 –> 00:09:41,630 lies inside the set.
350 00:09:41,640 –> 00:09:43,460 S. Here,
351 00:09:43,469 –> 00:09:45,030 Please recall, we need at
352 00:09:45,039 –> 00:09:46,820 least one such U with
353 00:09:46,830 –> 00:09:48,359 this property to have
354 00:09:48,369 –> 00:09:49,530 interior points.
355 00:09:50,530 –> 00:09:52,080 In other words, we can’t
356 00:09:52,090 –> 00:09:53,669 fit an unbounded interval
357 00:09:53,679 –> 00:09:55,250 into this interval here.
358 00:09:56,090 –> 00:09:57,929 And therefore, the interior
359 00:09:57,940 –> 00:09:59,900 of S is the empty set.
360 00:10:00,739 –> 00:10:01,270 OK.
361 00:10:01,369 –> 00:10:03,030 Now you might ask, what can
362 00:10:03,039 –> 00:10:04,520 we say about the exterior
363 00:10:04,530 –> 00:10:05,039 points?
364 00:10:05,890 –> 00:10:07,250 And there, you know, we have
365 00:10:07,260 –> 00:10:08,789 to look at X without
366 00:10:08,799 –> 00:10:09,309 S.
367 00:10:10,119 –> 00:10:11,919 So the complement of S which
368 00:10:11,929 –> 00:10:13,650 is simply the interval for
369 00:10:13,659 –> 00:10:15,030 minus infinity
370 00:10:15,559 –> 00:10:17,369 to zero and then the
371 00:10:17,380 –> 00:10:19,030 union of the interval
372 00:10:19,039 –> 00:10:20,830 from one to infinity.
373 00:10:22,130 –> 00:10:23,780 So there you see this is
374 00:10:23,789 –> 00:10:25,419 better for this set.
375 00:10:25,429 –> 00:10:27,380 We find open sets that
376 00:10:27,390 –> 00:10:28,869 are contained in this one
377 00:10:29,729 –> 00:10:31,349 more concretely, this works
378 00:10:31,359 –> 00:10:32,940 for all points except
379 00:10:32,950 –> 00:10:34,849 one here in the second part.
380 00:10:35,869 –> 00:10:37,650 Therefore, the exterior of
381 00:10:37,659 –> 00:10:39,030 S is simply the
382 00:10:39,039 –> 00:10:40,570 interval from one to
383 00:10:40,580 –> 00:10:41,330 infinity.
384 00:10:42,390 –> 00:10:42,890 OK.
385 00:10:42,900 –> 00:10:43,929 Now, with the knowledge of
386 00:10:43,940 –> 00:10:45,489 these two sets, we
387 00:10:45,500 –> 00:10:46,869 immediately know all the
388 00:10:46,880 –> 00:10:48,400 boundary points of S
389 00:10:49,200 –> 00:10:49,609 there.
390 00:10:49,619 –> 00:10:51,320 Please recall these are all
391 00:10:51,330 –> 00:10:52,690 the points in X that are
392 00:10:52,700 –> 00:10:54,289 neither in the interior of
393 00:10:54,299 –> 00:10:56,250 S nor in the exterior
394 00:10:56,260 –> 00:10:56,969 of S.
395 00:10:57,669 –> 00:10:59,409 Hence, it’s the interval
396 00:10:59,419 –> 00:11:01,359 from minus infinity to
397 00:11:01,369 –> 00:11:02,770 one including the
398 00:11:02,780 –> 00:11:03,489 point 1.
399 00:11:04,349 –> 00:11:05,820 Again, this might look
400 00:11:05,830 –> 00:11:07,380 strange, but it is simply
401 00:11:07,390 –> 00:11:08,950 because our topology is
402 00:11:08,960 –> 00:11:10,130 chosen in this way,
403 00:11:10,929 –> 00:11:12,530 all open sets stretch from
404 00:11:12,539 –> 00:11:13,969 one point to plus
405 00:11:13,979 –> 00:11:14,799 infinity.
406 00:11:14,869 –> 00:11:16,580 So they can’t see what happens
407 00:11:16,590 –> 00:11:17,770 in the other direction to
408 00:11:17,780 –> 00:11:18,840 minus infinity.
409 00:11:19,640 –> 00:11:20,940 And therefore we have the
410 00:11:20,950 –> 00:11:22,659 conclusion that the boundary
411 00:11:22,669 –> 00:11:24,500 of this set is everything
412 00:11:24,510 –> 00:11:25,659 on the left hand side.
413 00:11:26,659 –> 00:11:27,109 OK.
414 00:11:27,119 –> 00:11:28,369 And in order to close this
415 00:11:28,380 –> 00:11:29,890 example here, now we can
416 00:11:29,900 –> 00:11:31,169 also say that the closure
417 00:11:31,179 –> 00:11:33,080 of S is exactly the same
418 00:11:33,090 –> 00:11:33,450 set.
419 00:11:34,400 –> 00:11:35,919 So you see this here is a
420 00:11:35,929 –> 00:11:37,750 nice example in order to
421 00:11:37,760 –> 00:11:39,380 get used to topologies
422 00:11:40,219 –> 00:11:41,950 and please never forget all
423 00:11:41,960 –> 00:11:43,369 the notions we have here
424 00:11:43,380 –> 00:11:45,070 always depend on the chosen
425 00:11:45,080 –> 00:11:45,799 topology.
426 00:11:46,809 –> 00:11:48,460 Then I would say let’s go
427 00:11:48,469 –> 00:11:49,900 deeper into the field with
428 00:11:49,909 –> 00:11:51,099 the next videos.
429 00:11:51,869 –> 00:11:53,169 Therefore, I hope I see you
430 00:11:53,179 –> 00:11:54,679 there and have a nice day.
431 00:11:54,750 –> 00:11:55,520 Bye.
-
Quiz Content
Q1: Let $X$ be a set and $\mathcal{T}$ be a topology on $X$. What is the correct definition for $p$ being an interior point of $S \subseteq X$.
A1: There is a set $U \in \mathcal{T}$ with $U \subseteq X$.
A2: There is a set $U \in \mathcal{T}$ with $p \in U$ and $U \subseteq X$.
A3: There is a set $U \in \mathcal{T}$ with $p \in U$ and $U \subseteq \mathcal{T}$.
A4: There is a set $U \in \mathcal{T}$ with $p \in U$ and $U \subseteq S$.
A5: There is a set $U \in \mathcal{T}$ with $p \notin U$ and $U \subseteq X$.
Q2: Let $X$ be a set and $\mathcal{T}$ be a topology on $X$. What is not a correct definition for $p$ being an exterior point of $S \subseteq X$.
A1: $p$ is an interior point for $X \setminus S$.
A2: There is a set $U \in \mathcal{T}$ with $p \in U$ and $U \subseteq X \setminus S$.
A3: There is a set $U \in \mathcal{T}$ with $p \notin U$ and $U \subseteq X$.
Q3: Let $X=\mathbb{R}$ and $$\mathcal{T} = { \emptyset, \mathbb{R} } \cup { (a, \infty) \mid a \in \mathbb{R} }$$ be a topology on $\mathbb{R}$. What is $S^\prime$ for $S = (0,1)$.
A1: $S^\prime = (-\infty, 1]$.
A2: $S^\prime = (0, 1]$.
A3: $S^\prime = [0, 1]$.
A4: $S^\prime = (0, 1)$.
-
Last update: 2024-10
