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Is the concept of compact spaces a generalization of completeness to non-metric topological spaces?

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Not quite. A metric space space can be complete without being compact (e.g., $\mathbb R$ with the Euclidean topology). For a metric space, completeness + total boundedness = compactness.

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  • $\begingroup$ You could argue that local compactness is the generalization, but that's not quite true either. The Baire space is not locally compact, but it is complete. $\endgroup$
    – Asaf Karagila
    Sep 23, 2014 at 10:00
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Generalization of completeness to non-metric spaces goes through the concept of uniform spaces.

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