Require Export TopologicalSpaces.
Require Export Continuity.
Inductive homeomorphism {X Y:TopologicalSpace}
(f:point_set X -> point_set Y) : Prop :=
| intro_homeomorphism: forall g:point_set Y -> point_set X,
continuous f -> continuous g ->
(forall x:point_set X, g (f x) = x) ->
(forall y:point_set Y, f (g y) = y) -> homeomorphism f.

(* Specifications: *)
Lemma homeomorphism_is_invertible: forall {X Y:TopologicalSpace}
(f:point_set X -> point_set Y),
homeomorphism f -> invertible f.