Proof That Any Open Subspace of a Separable Space is Separable
Any open subspace of a separable space is separable.
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.