End product_topology.
Arguments ProductTopology {A}.
Arguments product_space_proj {A} {X}.

(* Specifications: *)
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).