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).

Lemma subspace_topology_topology: forall U:Ensemble {x:point_set X | In A x}, @open SubspaceTopology U -> exists V:Ensemble (point_set X), open V /\ U = inverse_image subspace_inc V.