Require Export Ensembles.
From ZornsLemma Require Export Families.
From ZornsLemma Require Export IndexedFamilies.
From ZornsLemma Require Export FiniteTypes.
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
}.
Arguments open {t}.
Arguments open_family_union {t}.
Arguments open_intersection2 {t}.
Definition closed {X:TopologicalSpace} (F:Ensemble X) :=
open (Ensembles.Complement F).

Lemma closed_finite_indexed_union: forall {X:TopologicalSpace}
{A:Type} (F:IndexedFamily A X),
FiniteT A -> (forall a:A, closed (F a)) ->
closed (IndexedUnion F).