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
}.

Lemma open_union2: forall {X:TopologicalSpace} (U V:Ensemble (point_set X)), open U -> open V -> open (Union U V).