Section SumTopology.
Variable A:Type.
Variable X:forall a:A, TopologicalSpace.
Inductive sum_space_point_set : Type :=
sum_space_point_set_intro (a : A) : point_set (X a) -> sum_space_point_set.
Definition SumTopology : TopologicalSpace :=
StrongTopology sum_space_point_set_intro.
End SumTopology.
Arguments sum_space_point_set {A}.
Arguments sum_space_point_set_intro {A}.

Lemma sum_topology_inj_image_open {A : Type} (X : forall a, TopologicalSpace) (a : A) (S : Ensemble (point_set (X a))) :
open S <-> @open (SumTopology A X) (Im S (sum_space_point_set_intro X a)).