Definition first_countable (X:TopologicalSpace) : Prop :=
  forall x:point_set X, exists NBx:Family (point_set X),
    neighborhood_basis NBx x /\ Countable NBx.

Lemma first_countable_open_neighborhood_bases: forall X:TopologicalSpace, first_countable X -> forall x:point_set X, exists NBx:Family (point_set X), open_neighborhood_basis NBx x /\ Countable NBx.