Definition neighborhood_filter {X:TopologicalSpace} (x0:point_set X) :
  Filter (point_set X).Definition filter_limit {X:TopologicalSpace} (F:Filter (point_set X))
  (x0:point_set X) : Prop :=
  Included (filter_family (neighborhood_filter x0))
           (filter_family F).Definition filter_cluster_point {X:TopologicalSpace}
  (F:Filter (point_set X)) (x0:point_set X) : Prop :=
  forall S:Ensemble (point_set X), In (filter_family F) S ->
  In (closure S) x0.

Lemma func_preserving_filter_limits_is_continuous: forall (X Y:TopologicalSpace) (f:point_set X -> point_set Y) (x:point_set X), (forall F:Filter (point_set X), filter_limit F x -> filter_limit (filter_direct_image f F) (f x)) -> continuous_at f x.