Definition Net := DS_set I -> point_set X.Definition net_cluster_point (x:Net) (x0:point_set X) : Prop :=
  forall U:Ensemble (point_set X), open U -> In U x0 ->
  exists arbitrarily large i:DS_set I, In U (x i).

(* Specifications: *)
Lemma net_limit_is_cluster_point: forall (x:Net) (x0:point_set X), net_limit x x0 -> net_cluster_point x x0.
Lemma net_limit_in_closure: forall (S:Ensemble (point_set X)) (x:Net) (x0:point_set X), (exists arbitrarily large i:DS_set I, In S (x i)) -> net_limit x x0 -> In (closure S) x0.