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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).

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What is the status of the extreme value theorem in forms of constructive mathematics, such a...

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 …
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