End Nets_and_continuity.
Section Subnet.
Variable X:TopologicalSpace.
Variable I:DirectedSet.
Variable x:Net I X.
Inductive Subnet {J:DirectedSet} : Net J X -> Prop :=
| intro_subnet: forall h:DS_set J -> DS_set I,
(forall j1 j2:DS_set J, DS_ord j1 j2 ->
DS_ord (h j1) (h j2)) ->
(exists arbitrarily large i:DS_set I,
exists j:DS_set J, h j = i) ->
Subnet (fun j:DS_set J => x (h j)).

Lemma subnet_cluster_point: forall (x0:point_set X) {J:DirectedSet}
(y:Net J X), net_cluster_point y x0 ->
Subnet y -> net_cluster_point x x0.