Theorem
Any open subspace of a separable space is separable.
Proof
Letbe a separable space and let
be a countable dense subset of
Ifis an open subset of
define
Letbe an open subset of
hence
is open in
hence
contains at least one
A subset of a topological spaceis dense in
if and only if every open subset of
contains a point of the subset. Therefore
is dense in
contains a countable dense subset
hence
is separable.