Definition dist_to_set (x:X) : R.

(* Specifications: *)
Lemma dist_to_set_triangle_inequality: forall (x y:X), dist_to_set y <= dist_to_set x + d x y.
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.
Lemma metrizable_impl_normal_sep: forall X:TopologicalSpace, metrizable X -> normal_sep X.