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_img: forall {X Y:TopologicalSpace}
(f:point_set X -> point_set Y),
connected X -> continuous f -> surjective f -> connected Y.