Is every closed subset of some ambient complete space complete? 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?
 A: 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.
