Definition ProductTopology2 : TopologicalSpace :=
  WeakTopology prod2_proj.

Lemma product2_map_continuous: forall (W:TopologicalSpace) (f:point_set W -> point_set X) (g:point_set W -> point_set Y) (w:point_set W), continuous_at f w -> continuous_at g w -> continuous_at (fun w:point_set W => (f w, g w)) w (Y:=ProductTopology2).