Section continuity.
Variable X Y:TopologicalSpace.
Variable f:point_set X -> point_set Y.
Definition continuous : Prop :=
forall V:Ensemble (point_set Y), open V ->
open (inverse_image f V).
Definition continuous_at (x:point_set X) : Prop :=
forall V:Ensemble (point_set Y),
neighborhood V (f x) -> neighborhood (inverse_image f V) x.
End continuity.
Arguments continuous {X} {Y}.
Arguments continuous_at {X} {Y}.

Lemma dense_image_surjective {X Y : TopologicalSpace} {f : point_set X -> point_set Y}
(S : Ensemble (point_set X)) :
continuous f ->
surjective f ->
dense S ->
dense (Im S f).