Record TopologicalSpace : Type := {
  point_set : Type;
  open : Ensemble point_set -> Prop;
  open_family_union : forall F : Family point_set,
    (forall S : Ensemble point_set, In F S -> open S) ->
    open (FamilyUnion F);
  open_intersection2: forall U V:Ensemble point_set,
    open U -> open V -> open (Intersection U V);
  open_full : open Full_set
}.Record TopologicalSpace : Type := {
  point_set : Type;
  open : Ensemble point_set -> Prop;
  open_family_union : forall F : Family point_set,
    (forall S : Ensemble point_set, In F S -> open S) ->
    open (FamilyUnion F);
  open_intersection2: forall U V:Ensemble point_set,
    open U -> open V -> open (Intersection U V);
  open_full : open Full_set
}.Record TopologicalSpace : Type := {
  point_set : Type;
  open : Ensemble point_set -> Prop;
  open_family_union : forall F : Family point_set,
    (forall S : Ensemble point_set, In F S -> open S) ->
    open (FamilyUnion F);
  open_intersection2: forall U V:Ensemble point_set,
    open U -> open V -> open (Intersection U V);
  open_full : open Full_set
}.Record TopologicalSpace : Type := {
  point_set : Type;
  open : Ensemble point_set -> Prop;
  open_family_union : forall F : Family point_set,
    (forall S : Ensemble point_set, In F S -> open S) ->
    open (FamilyUnion F);
  open_intersection2: forall U V:Ensemble point_set,
    open U -> open V -> open (Intersection U V);
  open_full : open Full_set
}.Record TopologicalSpace : Type := {
  point_set : Type;
  open : Ensemble point_set -> Prop;
  open_family_union : forall F : Family point_set,
    (forall S : Ensemble point_set, In F S -> open S) ->
    open (FamilyUnion F);
  open_intersection2: forall U V:Ensemble point_set,
    open U -> open V -> open (Intersection U V);
  open_full : open Full_set
}.Record TopologicalSpace : Type := {
  point_set : Type;
  open : Ensemble point_set -> Prop;
  open_family_union : forall F : Family point_set,
    (forall S : Ensemble point_set, In F S -> open S) ->
    open (FamilyUnion F);
  open_intersection2: forall U V:Ensemble point_set,
    open U -> open V -> open (Intersection U V);
  open_full : open Full_set
}.

(* Specifications: *)
Lemma open_empty: forall X:TopologicalSpace, open (@Empty_set (point_set X)).
Lemma open_union2: forall {X:TopologicalSpace} (U V:Ensemble (point_set X)), open U -> open V -> open (Union U V).
Lemma open_indexed_union: forall {X:TopologicalSpace} {A:Type} (F:IndexedFamily A (point_set X)), (forall a:A, open (F a)) -> open (IndexedUnion F).
Lemma open_finite_indexed_intersection: forall {X:TopologicalSpace} {A:Type} (F:IndexedFamily A (point_set X)), FiniteT A -> (forall a:A, open (F a)) -> open (IndexedIntersection F).