WEBVTT 00:00:00.250 --> 00:00:04.790 A function assigns exactly one output point to every input point. 00:00:05.256 --> 00:00:10.400 To study its interaction with observation, also ask about target events: which 00:00:10.410 --> 00:00:14.104 inputs land inside a specified subset of the target? 00:00:14.547 --> 00:00:17.444 This question runs backward through the point arrows. 00:00:18.564 --> 00:00:21.615 That set of inputs is the inverse image of U. 00:00:22.187 --> 00:00:24.152 It is not an inverse function. 00:00:24.624 --> 00:00:28.661 Several inputs may have the same output, and the inverse image is 00:00:28.661 --> 00:00:32.372 still defined even when no inverse function exists. 00:00:33.536 --> 00:00:37.477 On a two-point target, let every input map to one. 00:00:37.956 --> 00:00:42.699 The inverse image of the singleton containing one is the whole source. 00:00:43.135 --> 00:00:46.561 The inverse image of an event missing one is empty. 00:00:47.044 --> 00:00:50.493 Both sets are open in any source topology. 00:00:51.356 --> 00:00:55.935 A function is continuous when the inverse image of every open target 00:00:55.935 --> 00:00:58.236 event is open in the source. 00:00:58.672 --> 00:01:04.826 This definition concerns events, not whether the arrows look smooth or endpoints 00:01:04.847 --> 00:01:06.933 look near each other in a drawing. 00:01:08.025 --> 00:01:11.899 Use separate two-point source and target spaces. 00:01:12.548 --> 00:01:17.558 Each has opens empty, the singleton one, and the whole two-point 00:01:17.569 --> 00:01:21.201 carrier; the singleton zero is not open. 00:01:21.684 --> 00:01:26.710 Consider negation as a point function, exchanging zero and one. 00:01:27.189 --> 00:01:32.284 Test continuity by pulling back the target open singleton one. 00:01:33.194 --> 00:01:37.447 Under negation, only input zero lands at one. 00:01:37.959 --> 00:01:43.221 The inverse image is the singleton containing zero, which is not source- 00:01:43.221 --> 00:01:43.717 open. 00:01:44.200 --> 00:01:50.353 One failed target-open test disproves continuity; the map's bijectivity does 00:01:50.353 --> 00:01:52.160 not repair that failure. 00:01:53.015 --> 00:01:57.854 A selected formula is a true-or-false question on valuations. 00:01:58.479 --> 00:02:02.589 Its true-row region is the set where the answer is one. 00:02:03.083 --> 00:02:09.398 These regions generate the observation opens using unions and finite intersections, together 00:02:09.398 --> 00:02:11.155 with the empty and whole sets. 00:02:12.320 --> 00:02:17.789 Each selected formula defines a bit-valued function: it sends a valuation 00:02:17.821 --> 00:02:20.025 to that formula's truth value. 00:02:20.515 --> 00:02:24.005 Give its two-point target the Sierpinski topology. 00:02:24.569 --> 00:02:28.910 The inverse image of the target open singleton one is exactly the 00:02:28.910 --> 00:02:31.148 selected formula's truth region. 00:02:32.125 --> 00:02:37.839 That truth region is a designated subbasic open in the observation topology. 00:02:38.390 --> 00:02:42.651 The other target opens pull back to empty and the whole carrier. 00:02:43.112 --> 00:02:48.343 Thus every selected coordinate map is continuous by direct checking of all 00:02:48.353 --> 00:02:49.395 target opens. 00:02:50.515 --> 00:02:55.180 It is enough to check inverse images of target subbasic opens. 00:02:55.815 --> 00:03:02.180 Inverse images preserve intersections and unions, so once those generators pull back 00:03:02.180 --> 00:03:07.593 to opens, all their finite intersections and arbitrary unions do too. 00:03:08.633 --> 00:03:13.516 Continuity is a statement about a function together with two topologies. 00:03:14.121 --> 00:03:20.327 Changing either observation menu can change whether the inverse-image test passes. 00:03:20.987 --> 00:03:25.403 A point-arrow diagram without its source and target opens does not 00:03:25.403 --> 00:03:27.027 establish continuity. 00:03:28.078 --> 00:03:33.139 On the positive two-point spaces, compare the constant-one map with 00:03:33.139 --> 00:03:33.893 negation. 00:03:34.389 --> 00:03:38.643 What is the inverse image of the target singleton one for each, 00:03:38.967 --> 00:03:41.061 and which map is continuous? 00:03:59.398 --> 00:04:03.631 The constant-one preimage is the whole source, which is open. 00:04:04.121 --> 00:04:07.687 Negation gives the non-open singleton zero. 00:04:08.165 --> 00:04:14.296 The constant map passes all open tests and is continuous; negation fails 00:04:14.318 --> 00:04:16.004 this one and is not. 00:04:16.984 --> 00:04:21.596 Inverse images translate target events into source conditions. 00:04:22.236 --> 00:04:29.354 Continuity says every observable open target condition remains observable after that translation. 00:04:29.896 --> 00:04:34.700 Next, the finite information orders will let us check the same property 00:04:34.721 --> 00:04:37.065 through preservation of comparisons.