Section metric_topology.
Variable X:Type.
Variable d:X->X->R.
Hypothesis d_is_metric: metric d.
Definition open_ball (x0:X) (r:R) : Ensemble X :=
[ x:X | d x0 x < r ].
Inductive metric_topology_neighborhood_basis (x:X) : Family X :=
| intro_open_ball: forall r:R, r > 0 ->
In (metric_topology_neighborhood_basis x) (open_ball x r).
Definition MetricTopology : TopologicalSpace.
refine (Build_TopologicalSpace_from_open_neighborhood_bases
X metric_topology_neighborhood_basis _ _ _ _);
intros.
- destruct H as [r1 ?].
destruct H0 as [r2 ?].
exists (open_ball x (Rmin r1 r2)); split.
+ constructor.
apply Rmin_Rgt_r; split; trivial.
+ red. intros.
destruct H1.
constructor;
constructor;
apply Rlt_le_trans with (Rmin r1 r2); trivial.
* apply Rmin_l.
* apply Rmin_r.
- destruct H.
constructor.
now rewrite metric_zero.
- exists (open_ball x 1).
constructor.
auto with *.
- destruct H.
destruct H0.
exists (open_ball y (r - d x y)); split.
+ constructor.
now apply Rgt_minus.
+ red. intros z ?.
destruct H1.
constructor.
apply Rle_lt_trans with (d x y + d y z).
* now apply triangle_inequality.
* assert (d x y + d y z < d x y + (r - d x y)) by
auto with *.
now ring_simplify in H2.
Defined.
End metric_topology.
Arguments metric_topology_neighborhood_basis {X}.
Arguments MetricTopology {X}.
Definition metrizes (X:TopologicalSpace)
(d:point_set X -> point_set X -> R) : Prop :=
forall x:point_set X, open_neighborhood_basis
(metric_topology_neighborhood_basis d x) x.
Inductive metrizable (X:TopologicalSpace) : Prop :=
| intro_metrizable: forall d:point_set X -> point_set X -> R,
metric d -> metrizes X d ->
metrizable X.
Require Export Nets.

Lemma metric_space_net_cluster_point: forall (X:TopologicalSpace)
(d:point_set X -> point_set X -> R), metrizes X d ->
forall (I:DirectedSet) (x:Net I X) (x0:point_set X),
(forall eps:R, eps > 0 ->
exists arbitrarily large i:DS_set I, d x0 (x i) < eps) ->
net_cluster_point x x0.