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 R_compact_subset_bounded: forall A:Ensemble (point_set RTop),
compact (SubspaceTopology A) -> bound A.