Require Export TopologicalSpaces OrderTopology Completeness MetricSpaces.
Require Export Compactness Connectedness.
Definition RTop := OrderTopology Rle.
Definition R_metric (x y:R) : R := Rabs (y-x).

Lemma bounded_real_net_has_cluster_point: forall (I:DirectedSet)
(x:Net I RTop) (a b:R), (forall i:DS_set I, a <= x i <= b) ->
exists x0:point_set RTop, net_cluster_point x x0.