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 RTop_neighborhood_is_neighbourhood
(U : Ensemble R)
(r : R) :
@neighborhood RTop U r <-> neighbourhood U r.