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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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Is there a universal property characterizing the category of compact Hausdorff spaces?
We can describe $\mathbf{CHaus}$ with a universal property inside the $2$-category of all cocomplete categories and cocontinuous functors. Namely, $\mathbf{CHaus}$ is the universal cocomplete category …