Require Export TopologicalSpaces.
Require Export Continuity.
Require Export SubspaceTopology.
Section continuous_factorization.
Variable X Y:TopologicalSpace.
Variable f:point_set X -> point_set Y.
Variable S:Ensemble (point_set Y).
Hypothesis f_cont: continuous f.
Hypothesis f_img: forall x:point_set X, In S (f x).
Definition continuous_factorization :
point_set X -> point_set (SubspaceTopology S) :=
fun x:point_set X => exist _ (f x) (f_img x).

Lemma factorization_is_continuous:
continuous continuous_factorization.