Section dist_to_set_and_topology.
Variable X:TopologicalSpace.
Variable d:point_set X -> point_set X -> R.
Hypothesis d_is_metric: metric d.
Hypothesis d_metrizes_X: metrizes X d.
Variable S:Ensemble (point_set X).
Hypothesis S_nonempty: Inhabited S.
Variable T:Ensemble (point_set X).
Hypothesis T_nonempty: Inhabited T.
End dist_to_set_and_topology.

Lemma metrizable_impl_normal_sep: forall X:TopologicalSpace,
metrizable X -> normal_sep X.