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.

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: $S^\prime = (0, 1)$.