Record metric : Prop := {
  metric_nonneg: forall x y:X, d x y >= 0;
  metric_sym: forall x y:X, d x y = d y x;
  triangle_inequality: forall x y z:X, d x z <= d x y + d y z;
  metric_zero: forall x:X, d x x = 0;
  metric_strict: forall x y:X, d x y = 0 -> x = y
}.Record metric : Prop := {
  metric_nonneg: forall x y:X, d x y >= 0;
  metric_sym: forall x y:X, d x y = d y x;
  triangle_inequality: forall x y z:X, d x z <= d x y + d y z;
  metric_zero: forall x:X, d x x = 0;
  metric_strict: forall x y:X, d x y = 0 -> x = y
}.Record metric : Prop := {
  metric_nonneg: forall x y:X, d x y >= 0;
  metric_sym: forall x y:X, d x y = d y x;
  triangle_inequality: forall x y z:X, d x z <= d x y + d y z;
  metric_zero: forall x:X, d x x = 0;
  metric_strict: forall x y:X, d x y = 0 -> x = y
}.Record metric : Prop := {
  metric_nonneg: forall x y:X, d x y >= 0;
  metric_sym: forall x y:X, d x y = d y x;
  triangle_inequality: forall x y z:X, d x z <= d x y + d y z;
  metric_zero: forall x:X, d x x = 0;
  metric_strict: forall x y:X, d x y = 0 -> x = y
}.Record metric : Prop := {
  metric_nonneg: forall x y:X, d x y >= 0;
  metric_sym: forall x y:X, d x y = d y x;
  triangle_inequality: forall x y z:X, d x z <= d x y + d y z;
  metric_zero: forall x:X, d x x = 0;
  metric_strict: forall x y:X, d x y = 0 -> x = y
}.Record metric : Prop := {
  metric_nonneg: forall x y:X, d x y >= 0;
  metric_sym: forall x y:X, d x y = d y x;
  triangle_inequality: forall x y z:X, d x z <= d x y + d y z;
  metric_zero: forall x:X, d x x = 0;
  metric_strict: forall x y:X, d x y = 0 -> x = y
}.Inductive metric_topology_neighborhood_basis (x:X) : Family X :=
  | intro_open_ball: forall r:R, r > 0 ->
    In (metric_topology_neighborhood_basis x) (open_ball x r).Inductive metric_topology_neighborhood_basis (x:X) : Family X :=
  | intro_open_ball: forall r:R, r > 0 ->
    In (metric_topology_neighborhood_basis x) (open_ball x r).Definition MetricTopology : TopologicalSpace.Definition metrizes (X:TopologicalSpace)
  (d:point_set X -> point_set X -> R) : Prop :=
  forall x:point_set X, open_neighborhood_basis
             (metric_topology_neighborhood_basis d x) x.Inductive metrizable (X:TopologicalSpace) : Prop :=
  | intro_metrizable: forall d:point_set X -> point_set X -> R,
    metric d -> metrizes X d ->
    metrizable X.Inductive metrizable (X:TopologicalSpace) : Prop :=
  | intro_metrizable: forall d:point_set X -> point_set X -> R,
    metric d -> metrizes X d ->
    metrizable X.

Lemma metric_space_fun_continuity_converse: forall (X Y:TopologicalSpace) (f:point_set X->point_set Y) (x:point_set X) (dX:point_set X -> point_set X -> R) (dY:point_set Y -> point_set Y -> R), metrizes X dX -> metrizes Y dY -> continuous_at f x -> forall eps:R, eps > 0 -> exists delta:R, delta > 0 /\ forall x':point_set X, dX x x' < delta -> dY (f x) (f x') < eps.