Require Export TopologicalSpaces.
Require Export WeakTopology.
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_space_proj_continuous: forall a:A,
continuous (product_space_proj a) (X:=ProductTopology).