Corollary RTop_metrizable: metrizable RTop.

(* Specifications: *)
Lemma RTop_separable: separable RTop.
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_compact_subset_bounded: forall A:Ensemble (point_set RTop), compact (SubspaceTopology A) -> bound A.
Lemma Ropp_continuous: continuous Ropp (X:=RTop) (Y:=RTop).
Lemma R_connected: connected RTop.
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.