Record subbasis : Prop := {
  subbasis_elements: forall U:Ensemble (point_set X),
    In SB U -> open U;
  subbasis_cover: forall (U:Ensemble (point_set X)) (x:point_set X),
    In U x -> open U ->
    exists A:Type, FiniteT A /\
    exists V:A->Ensemble (point_set X),
      (forall a:A, In SB (V a)) /\
      In (IndexedIntersection V) x /\
      Included (IndexedIntersection V) U
}.Record subbasis : Prop := {
  subbasis_elements: forall U:Ensemble (point_set X),
    In SB U -> open U;
  subbasis_cover: forall (U:Ensemble (point_set X)) (x:point_set X),
    In U x -> open U ->
    exists A:Type, FiniteT A /\
    exists V:A->Ensemble (point_set X),
      (forall a:A, In SB (V a)) /\
      In (IndexedIntersection V) x /\
      Included (IndexedIntersection V) U
}.Record subbasis : Prop := {
  subbasis_elements: forall U:Ensemble (point_set X),
    In SB U -> open U;
  subbasis_cover: forall (U:Ensemble (point_set X)) (x:point_set X),
    In U x -> open U ->
    exists A:Type, FiniteT A /\
    exists V:A->Ensemble (point_set X),
      (forall a:A, In SB (V a)) /\
      In (IndexedIntersection V) x /\
      Included (IndexedIntersection V) U
}.

Lemma open_basis_is_subbasis: open_basis SB -> subbasis.