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_net_limit: forall (I:DirectedSet) (x:Net I ProductTopology) (x0:point_set ProductTopology), inhabited (DS_set I) -> (forall a:A, net_limit (fun i:DS_set I => x i a) (x0 a)) -> net_limit x x0.