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Proof That a Separable Metric Space is Homeomorphic to a Dense Subspace of a Complete Metric Space

Theorem

A separable metric spaceis homeomorphic to a dense subspace of a complete metric space.

Proof

A space is said to be separable if it contains a countable dense subset.

Any separable metric spacehas a metrizable compactification,say. Ifis compact, then it is complete.