Section product_topology.
Variable A:Type.
Variable X:forall a:A, TopologicalSpace.
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.