Timeline for Duality between compactness and Hausdorffness
Current License: CC BY-SA 3.0
7 events
| when toggle format | what | by | license | comment | |
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| Nov 6, 2015 at 14:44 | comment | added | Andrej Bauer | @AndréHenriques: there are several approaches to computable topology, but yes, in most of them a computable space is a space with an extra computable function. For instance, you could pick an enumeration of basic opens (assuming we have countably-based spaces only) and that will induce computability on points and continuous maps. Another possibiity is that you represent the computable structure of a space $X$ by a continuous (partial) surjection $\mathbb{N}^\mathbb{N} \to X$, and this will induce computability on $X$. I can give references if desired. | |
| Nov 6, 2015 at 9:53 | comment | added | André Henriques | Excuse my ignorance (I've looked at a couple of references but could not figure out an answer): What are the objects of the category of "computable topological spaces"? Is it the case that a computable topological space is a topological space equipped with some extra structure? Is there a functor from computable topological spaces to topological spaces? Is a computable topological space some kind of algorithm, written in some language (in which case there would be presumably only countably many computable topological spaces)? | |
| Nov 6, 2015 at 7:04 | comment | added | Andrej Bauer | Non-trivial overt spaces appear in computable topology, for example, as there the map $\exists_X$ needs to be computable. For instance, take $X = (0,a)$, an open interval with Euclidean topology where $a < 1$ is a real number such that for rational $q$ it is not semi-decidable whether $q < a$. If this space were overt then we could semi-decide whether $q < a$ by calculating $\exists_X((q, 1) \cap X)$. | |
| Nov 6, 2015 at 0:03 | comment | added | Zhen Lin | Let $f : X \to S$ be a continuous map of Hausdorff spaces (or, more generally, locales). Then, following Grothendieck's relative point of view, we have a space "relative to $S$", and it is overt if and only if $f : X \to S$ is an open map. | |
| Nov 5, 2015 at 23:26 | comment | added | André Henriques | What's an example of something that's not overt? (I'm aware that in the category of topological spaces, everything is overt, which is why I say "something") | |
| Nov 5, 2015 at 23:25 | history | edited | Zhen Lin | CC BY-SA 3.0 |
added 4 characters in body
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| Nov 5, 2015 at 21:34 | history | answered | Andrej Bauer | CC BY-SA 3.0 |