End WeakTopology_and_Nets.
End WeakTopology.
Arguments WeakTopology {X} {A} {Y}.
Arguments weak_topology_subbasis {X} {A} {Y}.
Section WeakTopology1.
Variable X:Type.
Variable Y:TopologicalSpace.
Variable f:X->point_set Y.
Definition WeakTopology1 := WeakTopology (True_rect f).

Lemma weak_topology1_topology:
forall U:Ensemble X, @open WeakTopology1 U ->
exists V:Ensemble (point_set Y), open V /\ U = inverse_image f V.