End dist_to_set.
Arguments dist_to_set {X}.
Arguments dist_to_set_triangle_inequality {X}.
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.

(* Specifications: *)
Lemma dist_to_set_zero_impl_closure: forall x:point_set X,
dist_to_set d d_is_metric S S_nonempty x = 0 -> In (closure S) x.
Lemma closure_impl_dist_to_set_zero: forall x:point_set X,
In (closure S) x -> dist_to_set d d_is_metric S S_nonempty x = 0.