Require Export TopologicalSpaces Homeomorphisms SubspaceTopology.
Definition clopen {X:TopologicalSpace} (S:Ensemble (point_set X))
: Prop :=
open S /\ closed S.
Definition connected (X:TopologicalSpace) : Prop :=
forall S:Ensemble (point_set X), clopen S ->
S = Empty_set \/ S = Full_set.

Lemma connected_union: forall {X:TopologicalSpace}
{A:Type} (S:IndexedFamily A (point_set X)),
(forall a:A, connected (SubspaceTopology (S a))) ->
Inhabited (IndexedIntersection S) ->
IndexedUnion S = Full_set -> connected X.