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

(* Specifications: *)
Lemma R_metric_is_metric: metric R_metric.
Lemma Rmetric_bound: forall x y z:R, R_metric x z < y - x ->
z < y.
Lemma Rmetric_bound2: forall x y z:R, R_metric y z < y - x ->
x < z.
Lemma RTop_metrization: metrizes RTop R_metric.
Corollary RTop_metrizable: metrizable RTop.
Lemma RTop_separable: separable RTop.
Require Export Compactness.
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.
Lemma R_closed_interval_compact: forall a b:R, a <= b ->
compact (SubspaceTopology ([x:point_set RTop | a <= x <= b])).
Lemma R_compact_subset_bounded: forall A:Ensemble (point_set RTop),
compact (SubspaceTopology A) -> bound A.
Lemma Ropp_continuous: continuous Ropp (X:=RTop) (Y:=RTop).
Require Export Connectedness.
Lemma R_connected: connected RTop.
Require Export Completeness.
Lemma R_cauchy_sequence_bounded: forall x:nat->R,
cauchy R_metric x -> bound (Im Full_set x).
Lemma R_cauchy_sequence_lower_bound: forall x:nat->R,
cauchy R_metric x -> lower_bound (Im Full_set x).
Lemma R_metric_complete: complete R_metric R_metric_is_metric.