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.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.

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