Require Export TopologicalSpaces.
Require Export Neighborhoods.
From ZornsLemma Require Export InverseImage.
Require Export OpenBases.
Require Export NeighborhoodBases.
Require Export Subbases.
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.

Lemma continuous_at_neighborhood_basis:
forall (x:point_set X) (NB:Family (point_set Y)),
neighborhood_basis NB (f x) ->
(forall V:Ensemble (point_set Y),
In NB V -> neighborhood (inverse_image f V) x) ->
continuous_at x.