Section Hausdorff_and_nets.
Require Export Nets.

(* Specifications: *)
Lemma Hausdorff_impl_net_limit_unique:
forall {X:TopologicalSpace} {I:DirectedSet} (x:Net I X),
Hausdorff X -> uniqueness (net_limit x).
Lemma Hausdorff_impl_net_limit_is_unique_cluster_point:
forall {X:TopologicalSpace} {I:DirectedSet} (x:Net I X)
(x0:point_set X), Hausdorff X -> net_limit x x0 ->
forall y:point_set X, net_cluster_point x y -> y = x0.
Lemma net_limit_is_unique_cluster_point_impl_Hausdorff:
forall (X:TopologicalSpace),
(forall (I:DirectedSet) (x:Net I X) (x0 y:point_set X),
net_limit x x0 -> net_cluster_point x y ->
y = x0) -> Hausdorff X.
Lemma net_limit_uniqueness_impl_Hausdorff:
forall X:TopologicalSpace,
(forall (I:DirectedSet) (x:Net I X), uniqueness (net_limit x)) ->
Hausdorff X.