Definition product_space_point_set : Type :=
  forall a:A, point_set (X a).Definition product_space_proj (a:A) : product_space_point_set ->
                                      point_set (X a) :=
  fun (x:product_space_point_set) => x a.Definition ProductTopology : TopologicalSpace :=
  WeakTopology product_space_proj.

Lemma product_map_continuous: forall {A:Type} (X:TopologicalSpace) (Y:A->TopologicalSpace) (f:forall a:A, point_set X -> point_set (Y a)) (x:point_set X), (forall a:A, continuous_at (f a) x) -> continuous_at (fun x:point_set X => (fun a:A => f a x)) x (Y:=ProductTopology Y).