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Results tagged with overt-spaces
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user 2733
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).
13
votes
Intermediate value theorem on computable reals
Thanks first to Andrej for drawing attention to
my paper on the IVT,
and indeed for his contributions to the work itself.
This paper is the introduction to Abstract Stone Duality
(my theory of computa …
- 8,481
6
votes
Accepted
What does overtness mean for metric spaces?
David Roberts has rubbed the magic lamp and the genie appears!
Even though the notion of overtness does depend on the strength of the ambient logic,
I believe the question here is with the notion of m …
- 8,481
15
votes
Is there a universal property characterizing the category of compact Hausdorff spaces?
As has already been remarked, $\mathbf{CHaus}$ is monadic over $\mathbf{Set}$.
It is also a pretopos, meaning roughly that it has the finitary properties of
$\mathbf{Set}$.
Another more symmetrical wa …
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16
votes
A topological concept dual to compactness
Andrej Bauer has given an excellent answer showing how overtness is dual to compactness, but to understand this idea more deeply and convince classical mathematicians that there is something in it, we …
- 8,481
0
votes
Is the Intermediate Value Theorem strictly stronger than LLPO?
There is, in my view, a regrettable custom, amongst even the most eminent people, that I call unconstructive mathematics, namely taking some classical result verbatim and showing that this implies exc …
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-3
votes
Condition to guarantee that an inhabited and bounded set of reals has a supremum
There is a topological analogue of locatedness, called overtness.
First let's consider how we obtain the supremum $a$ of $S$ as a Dedekind Cut,
$$ (u\in U) \quad\equiv\quad a< u \quad\equiv\quad \fora …
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5
votes
What is the status of the extreme value theorem in forms of constructive mathematics, such a...
By the extreme value theorem I take it that you mean that
Every continuous real-valued function on a closed bounded interval is bounded and attains its bounds.
I am going to answer this in terms …
- 8,481