Inductive weak_topology_subbasis : Family X :=
  | intro_fa_inv_image: forall (a:A) (V:Ensemble (point_set (Y a))),
    open V -> In weak_topology_subbasis (inverse_image (f a) V).Inductive weak_topology_subbasis : Family X :=
  | intro_fa_inv_image: forall (a:A) (V:Ensemble (point_set (Y a))),
    open V -> In weak_topology_subbasis (inverse_image (f a) V).Definition WeakTopology : TopologicalSpace :=
  Build_TopologicalSpace_from_subbasis X weak_topology_subbasis.

Lemma weak_topology_is_weakest: forall (T':Family X) (H1:_) (H2:_) (H3:_), (forall a:A, continuous (f a) (X := Build_TopologicalSpace X T' H1 H2 H3)) -> forall U:Ensemble X, @open WeakTopology U -> T' U.