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topology.metric_space.metrizable_uniformity

Metrizable uniform spaces #

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In this file we prove that a uniform space with countably generated uniformity filter is pseudometrizable: there exists a pseudo_metric_space structure that generates the same uniformity. The proof follows Sergey Melikhov, Metrizable uniform spaces.

Main definitions #

Main statements #

Tags #

metrizable space, uniform space

noncomputable def pseudo_metric_space.of_prenndist {X : Type u_1} (d : X X nnreal) (dist_self : (x : X), d x x = 0) (dist_comm : (x y : X), d x y = d y x) :

The maximal pseudo metric space structure on X such that dist x y ≤ d x y for all x y, where d : X → X → ℝ≥0 is a function such that d x x = 0 and d x y = d y x for all x, y.

Equations
theorem pseudo_metric_space.dist_of_prenndist {X : Type u_1} (d : X X nnreal) (dist_self : (x : X), d x x = 0) (dist_comm : (x y : X), d x y = d y x) (x y : X) :
has_dist.dist x y = (l : list X), (list.zip_with d (x :: l) (l ++ [y])).sum
theorem pseudo_metric_space.dist_of_prenndist_le {X : Type u_1} (d : X X nnreal) (dist_self : (x : X), d x x = 0) (dist_comm : (x y : X), d x y = d y x) (x y : X) :
has_dist.dist x y (d x y)
theorem pseudo_metric_space.le_two_mul_dist_of_prenndist {X : Type u_1} (d : X X nnreal) (dist_self : (x : X), d x x = 0) (dist_comm : (x y : X), d x y = d y x) (hd : (x₁ x₂ x₃ x₄ : X), d x₁ x₄ 2 * linear_order.max (d x₁ x₂) (linear_order.max (d x₂ x₃) (d x₃ x₄))) (x y : X) :
(d x y) 2 * has_dist.dist x y

Consider a function d : X → X → ℝ≥0 such that d x x = 0 and d x y = d y x for all x, y. Let dist be the largest pseudometric distance such that dist x y ≤ d x y, see pseudo_metric_space.of_prenndist. Suppose that d satisfies the following triangle-like inequality: d x₁ x₄ ≤ 2 * max (d x₁ x₂, d x₂ x₃, d x₃ x₄). Then d x y ≤ 2 * dist x y for all x, y.

@[protected]

If X is a uniform space with countably generated uniformity filter, there exists a pseudo_metric_space structure compatible with the uniform_space structure. Use uniform_space.pseudo_metric_space or uniform_space.metric_space instead.

@[protected]

A pseudo_metric_space instance compatible with a given uniform_space structure.

Equations
@[protected]

A metric_space instance compatible with a given uniform_space structure.

Equations
@[protected, instance]

A uniform space with countably generated 𝓤 X is pseudo metrizable.

A T₀ uniform space with countably generated 𝓤 X is metrizable. This is not an instance to avoid loops.