Topological Completeness is Weakly Hereditary

Theorem

Let $T = \struct {S, \tau}$ be a topological space which is topologically complete.

Let $V \subseteq S$ be a closed subspace of $T$.


Then $V$ is also topologically complete.


That is, topological completeness is weakly hereditary.


Proof


This needs considerable tedious hard slog to complete it.
In particular: Use Subspace of Complete Metric Space is Closed iff Complete
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Sources

  • 1978: Lynn Arthur Steen and J. Arthur Seebach, Jr.: Counterexamples in Topology (2nd ed.) ... (previous) ... (next): Part $\text I$: Basic Definitions: Section $5$: Metric Spaces: Complete Metric Spaces
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