Section Subspace.
Variable X:TopologicalSpace.
Variable A:Ensemble (point_set X).
Definition SubspaceTopology : TopologicalSpace :=
WeakTopology1 (proj1_sig (P:=fun x:point_set X => In A x)).
Definition subspace_inc : point_set SubspaceTopology ->
point_set X :=
proj1_sig (P:=fun x:point_set X => In A x).
End Subspace.
Arguments SubspaceTopology {X}.
Arguments subspace_inc {X}.

(* If the subspace [F] is closed in [X], then its [subspace_inc] is a closed map. *)
Lemma subspace_inc_takes_closed_to_closed
(X : TopologicalSpace) (F:Ensemble (point_set X)) :
closed F ->
forall G:Ensemble (point_set (SubspaceTopology F)),
closed G -> closed (Im G (subspace_inc F)).