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Proof That Any Open Subspace of a Separable Space is Separable

Theorem

Any open subspace of a separable space is separable.

Proof

Letbe a separable space and letbe a countable dense subset of

Ifis an open subset ofdefine

Letbe an open subset ofhenceis open inhencecontains at least one

A subset of a topological spaceis dense inif and only if every open subset of contains a point of the subset. Thereforeis dense incontains a countable dense subsethenceis separable.