Questions tagged [overt-spaces]
Overtness is the lattice dual of compactness in various forms of constructive topology and analysis, where related ideas are also called "located" (constructive analysis), "recursively enumerable" (computable analysis), "open" (locale theory) or "positive" (formal topology).
8 questions
8 votes
2 answers
682 views
Condition to guarantee that an inhabited and bounded set of reals has a supremum
This question is about constructive mathematics (without Choice), such as in the internal logic of a topos with natural numbers object, or in IZF. The “reals” (and the symbol $\mathbb{R}$) refer to ...
- 38k
14 votes
3 answers
2k views
Is there a universal property characterizing the category of compact Hausdorff spaces?
This is in some sense a follow up to the question asked here Properties of the category of compact Hausdorff spaces To clarify: The category $\text{Prof}$ of profinite sets sits inside the category $\...
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12 votes
2 answers
701 views
Is the Intermediate Value Theorem strictly stronger than LLPO?
(The context is Intuitionistic ZF set theory, or HoTT, or the internal logic of a topos with a Natural Number Object. The real numbers here mean the Dedekind reals.) By LLPO, I mean the statement that ...
- 4,993
5 votes
4 answers
814 views
What does overtness mean for metric spaces?
Original question: For compact metric spaces, plenty of subtly different definitions converge to the same concept. Overtness can be viewed as a property dual to compactness. So is there a similar ...
- 2,019
57 votes
3 answers
4k views
Duality between compactness and Hausdorffness
Consider a non-empty set $X$ and its complete lattice of topologies (see also this thread). The discrete topology is Hausdorff. Every topology that is finer than a Hausdorff topology is also ...
- 1,823
5 votes
2 answers
1k views
What is the status of the extreme value theorem in forms of constructive mathematics, such as Smooth Infinitesimal Analysis?
In certain intuitionistic frameworks the extreme value theorem cannot be proved. Depending on the exact framework, counterexamples can be constructed as well; see for example pp. 294-295 in Troelstra,...
- 18k
25 votes
7 answers
4k views
A topological concept dual to compactness
We say that a subset A in a topological space X is anti-compact if every covering of A by closed sets has a finite subcover. Clearly if X is Hausdorff then all anti-compact subsets of X are finite. ...
- 351
24 votes
5 answers
3k views
Intermediate value theorem on computable reals
Wikipedia says that the intermediate value theorem "depends on (and is actually equivalent to) the completeness of the real numbers." It then offers a simple counterexample to the analogous ...
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