Theorem
The first countable property is a topological property.
Proof
Supposeand
are homeomorphic so that
Letbe a homeomorphism and let
be first countable. Let
and let
be an open subset of
containing
Sinceis a homeomorphism
is a point in
and
is an open set in
such that
is first countable hence
has a countable local base
An open set
exists such that
then
where
is open in