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.

Lemma closer_to_S_than_T_open: open
[x:point_set X | dist_to_set d d_is_metric S S_nonempty x <
dist_to_set d d_is_metric T T_nonempty x].