WEBVTT 00:00:00.250 --> 00:00:02.848 X means dot and Y means stripe. 00:00:03.258 --> 00:00:07.035 On the original four-card carrier, A has only a dot, B 00:00:07.132 --> 00:00:10.077 both, C only a stripe, and D neither. 00:00:10.550 --> 00:00:14.879 A topology contains the empty and whole sets and stays closed under 00:00:14.933 --> 00:00:17.093 unions and finite intersections. 00:00:17.703 --> 00:00:22.987 With positive certificates, not receiving a certificate does not establish the opposite 00:00:22.987 --> 00:00:23.696 statement. 00:00:24.720 --> 00:00:30.442 Suppose the interface offers two positive certificates: X, true at A and 00:00:30.464 --> 00:00:32.894 B, and Y, true at B and C. 00:00:33.678 --> 00:00:36.500 Their truth regions are our starting subsets. 00:00:37.149 --> 00:00:41.501 We must find the smallest topology containing them on the unchanged four- 00:00:41.501 --> 00:00:42.565 state carrier. 00:00:43.500 --> 00:00:47.337 Every topology includes the empty set and the whole carrier. 00:00:47.878 --> 00:00:49.294 Add those immediately. 00:00:49.879 --> 00:00:54.034 Our starting family now has four members, but having the endpoints and 00:00:54.034 --> 00:00:58.993 the generators does not yet guarantee closure under the topology operations. 00:01:00.127 --> 00:01:03.947 Requiring both certificates means intersecting their regions. 00:01:04.596 --> 00:01:07.537 Only B satisfies both requirements. 00:01:08.192 --> 00:01:13.834 The singleton containing B must therefore be open in any topology containing 00:01:13.866 --> 00:01:15.364 the two starting regions. 00:01:16.488 --> 00:01:20.459 We have obtained one new open, not a new state. 00:01:20.953 --> 00:01:25.344 A and C remain in the carrier, although they fail this conjunction. 00:01:25.943 --> 00:01:31.639 D also remains possible until an actual premise is learned; describing available 00:01:31.639 --> 00:01:34.198 tests alone removes no point. 00:01:35.061 --> 00:01:38.007 Allowing either certificate gives the union. 00:01:38.576 --> 00:01:42.548 A is admitted by X, C by Y, and B by both. 00:01:43.177 --> 00:01:46.345 This creates the set containing A, B and C. 00:01:46.994 --> 00:01:51.197 Inclusive OR keeps the overlap rather than discarding it. 00:01:51.947 --> 00:01:55.051 Together with the endpoints, these are six opens. 00:01:55.684 --> 00:02:00.520 Further unions and finite intersections produce no new subset. 00:02:01.045 --> 00:02:04.926 This shows that the list is a topology, rather than merely a 00:02:04.926 --> 00:02:07.118 collection we happened to write down. 00:02:08.223 --> 00:02:14.513 Any topology containing the generators must contain their intersection, their union, and 00:02:14.513 --> 00:02:15.490 the endpoints. 00:02:16.129 --> 00:02:21.099 We have also checked that these forced sets already form a topology. 00:02:21.698 --> 00:02:26.624 It follows that no smaller legal topology can contain the two generators. 00:02:27.788 --> 00:02:31.109 The generating subsets are called a subbasis. 00:02:31.778 --> 00:02:37.659 Their finite intersections form a basis: the local pieces from which every 00:02:37.714 --> 00:02:39.168 open is a union. 00:02:39.807 --> 00:02:44.676 Here those pieces are the whole carrier, A and B, B and 00:02:44.698 --> 00:02:46.599 C, and B alone. 00:02:47.034 --> 00:02:53.440 The empty intersection supplies the whole carrier, including when no selected test 00:02:53.440 --> 00:02:54.540 is required. 00:02:55.451 --> 00:03:00.953 A complete minterm specifies a truth value for every variable. 00:03:01.388 --> 00:03:06.039 On our table, X together with not Y is the complete description 00:03:06.092 --> 00:03:06.671 of A. 00:03:07.725 --> 00:03:11.984 The minterm X and not Y defines A classically. 00:03:12.459 --> 00:03:16.838 But not Y is not an available test in this positive menu. 00:03:17.345 --> 00:03:23.549 Every generated open containing A also contains B, so the singleton containing 00:03:23.570 --> 00:03:25.866 A is absent from the topology. 00:03:26.726 --> 00:03:31.696 The singleton minterms from Lecture 05 partition all valuations. 00:03:32.200 --> 00:03:33.987 They are logical atoms. 00:03:34.575 --> 00:03:40.217 Calling them a topological basis here would incorrectly make every singleton open 00:03:40.238 --> 00:03:44.611 and erase the asymmetry our interface is supposed to express. 00:03:45.757 --> 00:03:50.568 Someone lists only the two generators, the empty set and the whole 00:03:50.568 --> 00:03:51.127 carrier. 00:03:51.678 --> 00:03:53.633 Is that already a topology? 00:03:54.206 --> 00:03:59.425 Name one missing intersection and one missing union before checking the answer. 00:04:16.163 --> 00:04:21.248 The intersection of the generators is the singleton containing B, which is 00:04:21.270 --> 00:04:21.799 missing. 00:04:22.315 --> 00:04:25.643 That one failure already disproves the proposal. 00:04:26.115 --> 00:04:31.776 A topology must satisfy all the rules, not merely contain some useful 00:04:31.798 --> 00:04:32.373 events. 00:04:33.552 --> 00:04:38.270 Their union, the set containing A, B and C, is missing too. 00:04:38.757 --> 00:04:43.389 Adding both missing sets gives the six-open topology we constructed. 00:04:43.860 --> 00:04:48.159 A generator list and a topology are different objects, linked by a 00:04:48.159 --> 00:04:49.720 closure construction.