Definition strong_open (S:Ensemble Y) : Prop :=
  forall a:A, open (inverse_image (f a) S).Definition StrongTopology : TopologicalSpace.

(* Specifications: *)
Lemma strong_topology_makes_continuous_funcs: forall a:A, continuous (f a) (Y:=StrongTopology).
Lemma strong_topology_strongest: forall (T':Ensemble Y->Prop) (H1:_) (H2:_) (H3:_), (forall a:A, continuous (f a) (Y:=Build_TopologicalSpace Y T' H1 H2 H3)) -> forall V:Ensemble Y, T' V -> strong_open V.