WEBVTT 00:00:00.250 --> 00:00:04.913 R and S are premise regions inside the same ambient world space. 00:00:05.407 --> 00:00:09.184 For their inclusion maps, a matching pair has the same world in 00:00:09.194 --> 00:00:10.312 both coordinates. 00:00:10.984 --> 00:00:16.088 Such pairs are exactly v,v for worlds in the intersection, which 00:00:16.088 --> 00:00:16.848 we call T. 00:00:17.642 --> 00:00:23.060 Equality here is equality of valuations, even when different valuations have the 00:00:23.060 --> 00:00:25.096 same observed signature. 00:00:25.895 --> 00:00:30.853 The matching pairs from the previous lecture correspond to the intersection points. 00:00:31.502 --> 00:00:36.711 To identify the two objects as topological spaces, we must also compare 00:00:36.711 --> 00:00:37.513 their opens. 00:00:38.148 --> 00:00:41.863 The carrier alone does not determine a topology. 00:00:42.774 --> 00:00:47.081 A bijection pairs the points on the two sides one-to-one, 00:00:47.250 --> 00:00:48.554 leaving none out. 00:00:49.131 --> 00:00:54.285 An inverse image consists of source points landing in a chosen target 00:00:54.285 --> 00:00:54.808 event. 00:00:55.379 --> 00:01:01.779 Continuity requires every open target event to have an open inverse image. 00:01:02.642 --> 00:01:07.807 A homeomorphism is a bijection that is continuous in both directions. 00:01:08.412 --> 00:01:13.447 It preserves the topological structure, not merely the number of points. 00:01:14.106 --> 00:01:18.295 We will use the diagonal map from the intersection to its compatible 00:01:18.295 --> 00:01:18.913 pairs. 00:01:20.033 --> 00:01:25.866 A subspace R inherits regions R intersect U from ambient opens U. 00:01:26.406 --> 00:01:31.017 In a product, an open rectangle requires an open condition on each 00:01:31.027 --> 00:01:35.552 coordinate; unions of these rectangles give the product opens. 00:01:36.172 --> 00:01:40.679 A coordinate projection simply reads one component of the pair. 00:01:41.617 --> 00:01:47.400 Each truth region inherits its topology by intersection with ambient opens. 00:01:48.040 --> 00:01:51.273 The intersection region inherits the same way. 00:01:51.797 --> 00:01:56.767 On the matching-pair set, use the subspace topology from the product 00:01:56.778 --> 00:01:58.926 of the two premise subspaces. 00:02:00.050 --> 00:02:06.036 The diagonal map has two continuous inclusion coordinates, so it is continuous 00:02:06.057 --> 00:02:09.754 into the product and into the matching-pair subspace. 00:02:10.248 --> 00:02:16.489 Either coordinate projection restricted to the matching pairs is its continuous inverse. 00:02:17.000 --> 00:02:20.571 Thus the identification is a homeomorphism. 00:02:21.562 --> 00:02:27.115 For the separate illustration, use the original four X,Y values: A 00:02:27.169 --> 00:02:32.158 is one-zero, B one-one, C zero-one and D zero- 00:02:32.179 --> 00:02:32.625 zero. 00:02:33.230 --> 00:02:35.614 Select positive X and Y. 00:02:36.140 --> 00:02:41.103 We will call the illustrative subspace J, keeping T for the general 00:02:41.136 --> 00:02:43.675 intersection R intersect S. 00:02:44.295 --> 00:02:47.546 One means true and zero means false. 00:02:48.498 --> 00:02:54.109 For a separate subspace example, restrict the positive four-state topology to 00:02:54.163 --> 00:02:57.165 A and B, and call this subspace J. 00:02:57.894 --> 00:03:02.472 The ambient Y region meets J at B, so B alone must 00:03:02.472 --> 00:03:04.807 be open in the induced topology. 00:03:05.351 --> 00:03:08.595 The induced opens are empty, B and A and B. 00:03:09.874 --> 00:03:14.594 If we instead assign only empty and the whole subspace as opens, 00:03:14.626 --> 00:03:19.021 the inclusion into the ambient space is no longer continuous. 00:03:19.680 --> 00:03:24.498 The ambient open Y region pulls back to B, which that arbitrary 00:03:24.519 --> 00:03:27.231 replacement topology does not contain. 00:03:28.431 --> 00:03:33.733 Now return to the general intersection T, equal to R intersect S. 00:03:34.382 --> 00:03:39.547 The separate J example is finished; R and S are again arbitrary 00:03:39.569 --> 00:03:43.324 premise subspaces in the same ambient space. 00:03:44.180 --> 00:03:49.790 Suppose two continuous maps from another space agree after inclusion into the 00:03:49.802 --> 00:03:50.966 ambient carrier. 00:03:51.421 --> 00:03:55.840 At every input z their output points are the same valuation. 00:03:56.199 --> 00:04:01.545 That valuation belongs to both premise regions, hence to their intersection. 00:04:02.409 --> 00:04:06.719 Assign this common value as the map into the intersection. 00:04:07.154 --> 00:04:11.712 It is the only possible assignment that can reproduce both given maps. 00:04:12.351 --> 00:04:17.245 Projecting the corresponding matching pair gives back f and g, so the 00:04:17.256 --> 00:04:19.198 required diagram factors. 00:04:20.327 --> 00:04:24.809 An open of the intersection is an ambient open restricted to T. 00:04:25.518 --> 00:04:30.617 Its inverse image under the common-value map is the inverse image 00:04:30.671 --> 00:04:34.116 under f of the corresponding open in R. 00:04:34.685 --> 00:04:38.352 That set is open because f is continuous. 00:04:39.511 --> 00:04:45.469 Any other mediator reproducing the same included values must assign the same 00:04:45.479 --> 00:04:48.483 common valuation at every input. 00:04:48.944 --> 00:04:53.502 This proves uniqueness as well as existence and continuity. 00:04:54.101 --> 00:04:59.513 These three requirements are the universal property in this inclusion case. 00:05:00.225 --> 00:05:04.819 For this check, take J to be the subspace containing A and 00:05:04.840 --> 00:05:04.992 B. 00:05:05.756 --> 00:05:10.284 The ambient Y region contains B and C and is open. 00:05:10.808 --> 00:05:14.845 What is its inverse image under the inclusion of J into the 00:05:14.856 --> 00:05:16.182 ambient space? 00:05:16.617 --> 00:05:21.120 Why would giving J only the empty and whole opens make that 00:05:21.120 --> 00:05:23.193 inclusion fail continuity? 00:05:41.464 --> 00:05:44.036 The required inverse image is B. 00:05:44.730 --> 00:05:49.515 It is open in the induced subspace topology but absent from the 00:05:49.535 --> 00:05:51.252 indiscrete replacement. 00:05:51.688 --> 00:05:57.278 The inclusion would fail continuity, so that replacement cannot simply stand in 00:05:57.332 --> 00:05:59.471 for the topological pullback. 00:06:00.394 --> 00:06:06.374 The pullback of truth-subspace inclusions is their intersection with its induced 00:06:06.438 --> 00:06:07.484 topology. 00:06:07.939 --> 00:06:12.982 The universal property records a unique continuous common-value map. 00:06:13.401 --> 00:06:19.336 It packages ordinary conjunction structurally; it does not alter classical validity or 00:06:19.390 --> 00:06:22.646 license arbitrary topologies on truth regions.