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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?
I definitely expect that there is much more than one good answer. But, here is one that one can get easily by just patching together several classical facts:
The category of compact Hausdorf topolog …