Definition UniformTopology : TopologicalSpace :=
  MetricTopology uniform_metric uniform_metric_is_metric.

(* Specifications: *)
Lemma uniform_metric_complete: complete d d_metric -> complete uniform_metric uniform_metric_is_metric.
Lemma continuous_functions_at_point_closed_in_uniform_metric: forall x0:point_set X, closed (fun f:uniform_space d y0 => continuous_at (proj1_sig f) x0 (Y:=MetricTopology d d_metric)) (X:=UniformTopology d y0 d_metric X_inh).
Lemma continuous_functions_closed_in_uniform_metric: closed (fun f:uniform_space d y0 => continuous (proj1_sig f) (Y:=MetricTopology d d_metric)) (X:=UniformTopology d y0 d_metric X_inh).