End UniformTopology.
Arguments uniform_space {X} {Y}.
Arguments uniform_metric {X} {Y}.
Arguments UniformTopology {X} {Y}.
Section UniformTopology_and_continuity.
Variable X:TopologicalSpace.
Variable Y:Type.
Variable d:Y->Y->R.
Variable y0:point_set X -> Y.
Hypothesis d_metric: metric d.
Hypothesis X_inh: inhabited (point_set X).

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).