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https://math.stackexchange.com/questions/244661/showing-that-if-a-subset-of-a-complete-metric-space-is-closed-it-is-also-comple#:~:text=If%20A%20is%20a%20closed,Thus%20A%20is%20complete.

We know that if X is complete and its subset A is closed, then A is complete in a metric space. I assume this holds true in non-metric space because in the link above the proof didn't seem to use the fact that X is metric space. But I can't see the official theorem for this in any source without the mention of metric space. Is this really true?

able20
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    There are some notions of completeness for non-metric spaces, but the notion of completeness under discussion in that question and its answers is a specifically metric notion, and the arguments there apply specifically to metric spaces. – Brian M. Scott Jan 31 '21 at 05:35

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As to the linked question: The fact that you're even using the term complete (and Cauchy sequences) implies that we're working in a metric space and "metric complete" is the being notion used. The titel is a strong hint. Plus the fact that closedness can be decided by convergent sequences is already using the metric implicitly. So the proofs there do use (part of) the metric.

It's also possible to define completeness in any uniform space, which is a different kind of struture from a topology,though related, as any uniform structure induces a topology (like a metric does). For that notion of completeness the result (a closed subset of a complete structure is again complete ) is also true, and might be the general fact you've seen. A complete metric space has a complete uniform structure.

There also is a notion of being "topologically complete", which is also used sometimes, and is also preserved by closed subsets, and is a class of spaces in which Baire's theorem holds, and also includes locally compact Hausdorff spaces, as well as completely metrisable spaces.

Henno Brandsma
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  • To the proposer: See the chapter "Uniform Spaces" in R. Engelking's "General Topology" if you can afford it. – DanielWainfleet Jan 31 '21 at 12:43
  • Thanks so much. Just as a follow-up, regarding the concept of convergence in the topological space, I thought it can be redefined as "there exists N such that if $n\geq N$ for some open neighborhood around p denoted as $N_p$, we have $a_n\in N_p$" but since we don't know the exact point $p$ for Cauchy sequence, we cannot redefine Cauchy sequence and accordingly completeness in topological space. Would it be the right way to think? – able20 Jan 31 '21 at 22:49
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    @able20 No, in general we need a concept of uniformity: a way to say that a pair of points is close or not. Or abstract it away as saying $X$ is a $G_\delta$ in its compactification, which is what the general topology notion is, but which looks totally unintuitive at first. – Henno Brandsma Jan 31 '21 at 22:52
  • Okay that makes sense. I'm still not sure what it means by the pair of points, because in topological space closed set is the complement of an open set. I might have to look it up. – able20 Jan 31 '21 at 22:59
  • @able20 Look up the entourage definition of uniformity. These are subsets of $X^2$ so pairs. – Henno Brandsma Jan 31 '21 at 23:08
  • Makes sense. Thank you! – able20 Jan 31 '21 at 23:09