Section UniformTopology.
Variable X Y:Type.
Variable d:Y->Y->R.
Variable y0:X->Y. (* base function limiting the extents of the
range of a function X->Y *)
Hypothesis d_metric: metric d.
Hypothesis X_inhabited: inhabited X.
Definition uniform_space := { f:X->Y | bound (Im Full_set
(fun x:X => d (y0 x) (f x))) }.
Definition uniform_metric (f g:uniform_space) : R.
refine (match f, g with exist f0 Hf, exist g0 Hg =>
proj1_sig (sup (Im Full_set (fun x:X => d (f0 x) (g0 x))) _ _) end).
- destruct Hf as [mf].
destruct Hg as [mg].
exists (mf + mg).
red; intros.
destruct H1.
rewrite H2.
apply Rle_trans with (d (y0 x) (f0 x) + d (y0 x) (g0 x)).
+ rewrite (metric_sym _ d d_metric (y0 x) (f0 x)); trivial;
apply triangle_inequality; trivial.
+ assert (d (y0 x) (f0 x) <= mf) by (apply H; exists x; trivial).
assert (d (y0 x) (g0 x) <= mg) by (apply H0; exists x; trivial).
auto with real.
- destruct X_inhabited as [x0].
exists (d (f0 x0) (g0 x0)); exists x0; trivial; constructor.
Defined.
Definition UniformTopology : TopologicalSpace :=
MetricTopology uniform_metric uniform_metric_is_metric.

Lemma uniform_metric_complete: complete d d_metric ->
complete uniform_metric uniform_metric_is_metric.