WEBVTT 00:00:00.250 --> 00:00:03.940 For this lecture, replace the four-card board with a separate carrier 00:00:03.950 --> 00:00:06.804 containing just two points, zero and one. 00:00:07.298 --> 00:00:10.810 A topology specifies which subsets count as open. 00:00:11.326 --> 00:00:15.665 It is a collection of subsets of this carrier, rather than one 00:00:15.665 --> 00:00:19.974 distinguished subset or a geometric arrangement of the two points. 00:00:21.103 --> 00:00:26.063 Consider the empty set, the singleton containing one, and the whole two- 00:00:26.095 --> 00:00:26.947 point set. 00:00:27.464 --> 00:00:30.351 We propose exactly these three opens. 00:00:30.975 --> 00:00:33.928 The singleton containing zero is absent. 00:00:34.459 --> 00:00:38.462 To check the proposal, we must test the topology rules. 00:00:39.592 --> 00:00:44.875 For the set notation, braces list members and the empty-set symbol 00:00:44.886 --> 00:00:46.167 means no members. 00:00:46.786 --> 00:00:49.518 Union collects members from either set. 00:00:50.073 --> 00:00:52.981 Intersection keeps only shared members. 00:00:53.630 --> 00:00:58.395 We are working on the separate two-point carrier, zero and one. 00:00:59.240 --> 00:01:02.865 The empty set and the whole carrier must be open. 00:01:03.377 --> 00:01:05.059 Our list includes both. 00:01:05.622 --> 00:01:11.124 Any union of listed opens must also be listed; unions in this 00:01:11.134 --> 00:01:16.591 example give only the empty set, the singleton containing one, or the 00:01:16.591 --> 00:01:17.624 whole carrier. 00:01:18.569 --> 00:01:23.834 Any finite intersection of listed opens must stay on the list. 00:01:24.426 --> 00:01:29.787 Intersecting the whole carrier with the singleton leaves that singleton; intersecting with 00:01:29.787 --> 00:01:32.194 the empty set gives the empty set. 00:01:32.740 --> 00:01:37.069 Repeating a set adds nothing, so the proposed family passes. 00:01:38.188 --> 00:01:44.134 Arbitrary unions allow any collection of open sets, including an infinite collection 00:01:44.134 --> 00:01:45.827 in a more general space. 00:01:46.409 --> 00:01:51.574 Finite intersections require only finitely many opens at once. 00:01:51.748 --> 00:01:56.848 These are different requirements, even though our current carrier has only finitely 00:01:56.858 --> 00:01:58.008 many subsets. 00:01:59.178 --> 00:02:03.398 On the real line, the usual open sets are unions of open 00:02:03.420 --> 00:02:03.952 balls. 00:02:04.683 --> 00:02:09.022 Small balls around a point fit inside such an open set. 00:02:09.548 --> 00:02:16.145 That familiar construction motivates topology, but a topology can also be specified 00:02:16.145 --> 00:02:19.445 directly, as our two-point example shows. 00:02:20.614 --> 00:02:24.965 We will use open neighborhood to mean an open set containing the 00:02:24.965 --> 00:02:26.154 point in question. 00:02:26.651 --> 00:02:30.591 The singleton containing one is an open neighborhood of one. 00:02:31.042 --> 00:02:35.176 The only open neighborhood of zero in our example is the whole 00:02:35.176 --> 00:02:35.926 carrier. 00:02:36.834 --> 00:02:41.629 An open neighborhood of one can exclude zero, but no open neighborhood 00:02:41.629 --> 00:02:43.481 of zero excludes one. 00:02:43.900 --> 00:02:46.646 Our definition permits this asymmetry. 00:02:46.689 --> 00:02:51.073 A topology is not required to make every singleton open or to 00:02:51.095 --> 00:02:53.669 come from an ordinary metric. 00:02:54.028 --> 00:02:57.834 A discrete topology declares every subset open. 00:02:58.319 --> 00:03:00.674 On two points it has four opens. 00:03:01.299 --> 00:03:06.355 An indiscrete topology declares only the empty set and the whole carrier 00:03:06.387 --> 00:03:06.865 open. 00:03:07.356 --> 00:03:11.002 Our three-open example lies between those choices. 00:03:12.127 --> 00:03:15.417 A set is closed when its complement is open. 00:03:16.009 --> 00:03:21.489 In the three-open example, the singleton containing zero is closed because 00:03:21.489 --> 00:03:25.274 its complement is the open singleton containing one. 00:03:25.800 --> 00:03:30.542 Closed does not mean empty of information or physically bounded by a 00:03:30.552 --> 00:03:31.180 line. 00:03:32.141 --> 00:03:36.386 The empty set and the whole carrier are both open and closed, 00:03:36.599 --> 00:03:37.536 or clopen. 00:03:38.081 --> 00:03:42.145 In a discrete topology every subset is clopen. 00:03:42.641 --> 00:03:46.906 A set can therefore be both open and closed; these words are 00:03:46.906 --> 00:03:48.859 not exclusive alternatives. 00:03:49.987 --> 00:03:55.119 Must the singleton containing zero be open because the singleton containing one 00:03:55.119 --> 00:03:55.804 is open? 00:03:56.316 --> 00:04:01.282 Check the topology rules, rather than appealing to symmetry in the picture. 00:04:15.423 --> 00:04:15.785 No. 00:04:16.272 --> 00:04:21.451 The three listed sets already satisfy the rules, and neither their unions 00:04:21.548 --> 00:04:25.496 nor their finite intersections produce the missing singleton. 00:04:26.022 --> 00:04:30.623 Taking complements is not a requirement for the collection of open sets. 00:04:31.738 --> 00:04:35.590 We have a legal topology with asymmetric neighborhoods. 00:04:36.273 --> 00:04:41.448 Next we will construct such families from selected subsets instead of being 00:04:41.470 --> 00:04:43.025 handed a finished list. 00:04:43.606 --> 00:04:48.369 Keep the generating subsets separate from the completed collection of opens.