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Results tagged with overt-spaces
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user 1176
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).
20
votes
Intermediate value theorem on computable reals
I am afraid Joel has missed an important detail there, which is worth pointing out. Suppose $f$ is continuous and computable on $[a,b]$ and $f(a) \cdot f(b) < 0$. We must be careful to distinguish bet …
- 4,713
65
votes
A topological concept dual to compactness
When looking for a dual concept we should be careful not to be tricked by a shallow symmetry. I will not comment on your definition of anti-compactness. Instead I would like to explain what the "true" …
- 4,713
4
votes
What does overtness mean for metric spaces?
To conjure up a non-overt space we must change slightly the definition of topology, since even inuitionistically every space is overt, so long as every union of opens is open.
Let $\Sigma$ be the Sier …
- 48.8k
17
votes
Duality between compactness and Hausdorffness
Hausdorff is dual to discrete. Compact is dual to overt.
A space $X$ is Hausdorff if and only if the diagonal $\Delta_X = \{(x,x) \mid x \in X\}$ is closed in $X \times X$. A space $X$ is discrete if …
- 15.9k