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.

(* Specifications: *)
Lemma connected_img: forall {X Y:TopologicalSpace} (f:point_set X -> point_set Y), connected X -> continuous f -> surjective f -> connected Y.
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.