Let convert_continuity: forall h:point_set X -> R,
continuous h (Y:=RTop) <-> continuous h (Y:=MetricTopology R_metric
R_metric_is_metric).

(* Specifications: *)
Lemma Tietze_extension_func_continuous: continuous Tietze_extension_func.
End Tietze_extension_construction.
Lemma bounded_Tietze_extension_theorem: forall (X:TopologicalSpace)
(F:Ensemble (point_set X)) (f:point_set (SubspaceTopology F) ->
point_set RTop),
normal_sep X -> closed F -> continuous f ->
(forall x:point_set (SubspaceTopology F), -1 <= f x <= 1) ->
exists g:point_set X -> point_set RTop,
continuous g /\ (forall x:point_set (SubspaceTopology F),
g (subspace_inc F x) = f x) /\
(forall x:point_set X, -1 <= g x <= 1).
Lemma open_bounded_Tietze_extension_theorem: forall (X:TopologicalSpace)
(F:Ensemble (point_set X)) (f:point_set (SubspaceTopology F) ->
point_set RTop),
normal_sep X -> closed F -> continuous f ->
(forall x:point_set (SubspaceTopology F), -1 < f x < 1) ->
exists g:point_set X -> point_set RTop,
continuous g /\ (forall x:point_set (SubspaceTopology F),
g (subspace_inc F x) = f x) /\
(forall x:point_set X, -1 < g x < 1).
Theorem Tietze_extension_theorem: forall (X:TopologicalSpace)
(F:Ensemble (point_set X)) (f:point_set (SubspaceTopology F) ->
point_set RTop),
normal_sep X -> closed F -> continuous f ->
exists g:point_set X -> point_set RTop,
continuous g /\ (forall x:point_set (SubspaceTopology F),
g (subspace_inc F x) = f x).