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_closed_interval_compact: forall a b:R, a <= b ->
compact (SubspaceTopology ([x:point_set RTop | a <= x <= b])).