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_filter_limit: forall (F:Filter (point_set ProductTopology)) (x0:point_set ProductTopology), (forall a:A, filter_limit (filter_direct_image (product_space_proj a) F) (x0 a)) -> filter_limit F x0.