Pseudo-metric spaces

This file defines pseudo-metric spaces: these differ from metric spaces by not imposing the condition dist x y = 0 → x = y. Many definitions and theorems expected on (pseudo-)metric spaces are already introduced on uniform spaces and topological spaces. For example: open and closed sets, compactness, completeness, continuity and uniform continuity.

Main definitions

• Dist α: Endows a space α with a function dist a b.
• PseudoMetricSpace α: A space endowed with a distance function, which can be zero even if the two elements are non-equal.
• Metric.ball x ε: The set of all points y with dist y x < ε.
• Metric.Bounded s: Whether a subset of a PseudoMetricSpace is bounded.
• MetricSpace α: A PseudoMetricSpace with the guarantee dist x y = 0 → x = y.

Additional useful definitions:

• nndist a b: dist as a function to the non-negative reals.
• Metric.closedBall x ε: The set of all points y with dist y x ≤ ε.
• Metric.sphere x ε: The set of all points y with dist y x = ε.

TODO (anyone): Add "Main results" section.

Tags

pseudo_metric, dist

theorem UniformSpace.ofDist_aux (ε : ℝ) (hε : 0 < ε) : ∃ δ > 0, ∀ x < δ, ∀ y < δ, x + y < ε

def UniformSpace.ofDist {α : Type u} (dist : α → α → ℝ) (dist_self : ∀ (x : α), dist x x = 0) (dist_comm : ∀ (x y : α), dist x y = dist y x) (dist_triangle : ∀ (x y z : α), dist x z ≤ dist x y + dist y z) : UniformSpace α

Construct a uniform structure from a distance function and metric space axioms

Equations

• UniformSpace.ofDist dist dist_self dist_comm dist_triangle = UniformSpace.ofFun dist dist_self dist_comm dist_triangle UniformSpace.ofDist_aux

@[reducible, inline]

abbrev Bornology.ofDist {α : Type u_3} (dist : α → α → ℝ) (dist_comm : ∀ (x y : α), dist x y = dist y x) (dist_triangle : ∀ (x y z : α), dist x z ≤ dist x y + dist y z) : Bornology α

Construct a bornology from a distance function and metric space axioms.

Equations

• Bornology.ofDist dist dist_comm dist_triangle = Bornology.ofBounded {s : Set α | ∃ (C : ℝ), ∀ ⦃x : α⦄, x ∈ s → ∀ ⦃y : α⦄, y ∈ s → dist x y ≤ C} ⋯ ⋯ ⋯ ⋯

theorem Dist.ext {α : Type u_3} (x : Dist α) (y : Dist α) (dist : Dist.dist = Dist.dist) : x = y

theorem Dist.ext_iff {α : Type u_3} (x : Dist α) (y : Dist α) : x = y ↔ dist = dist

class Dist (α : Type u_3) : Type u_3

The distance function (given an ambient metric space on α), which returns a nonnegative real number dist x y given x y : α.

• dist : α → α → ℝ

class PseudoMetricSpace (α : Type u) extends Dist : Type u

Pseudo metric and Metric spaces

A pseudo metric space is endowed with a distance for which the requirement d(x,y)=0 → x = y might not hold. A metric space is a pseudo metric space such that d(x,y)=0 → x = y. Each pseudo metric space induces a canonical UniformSpace and hence a canonical TopologicalSpace This is enforced in the type class definition, by extending the UniformSpace structure. When instantiating a PseudoMetricSpace structure, the uniformity fields are not necessary, they will be filled in by default. In the same way, each (pseudo) metric space induces a (pseudo) emetric space structure. It is included in the structure, but filled in by default.

• dist : α → α → ℝ
• dist_self : ∀ (x : α), dist x x = 0
• dist_comm : ∀ (x y : α), dist x y = dist y x
• dist_triangle : ∀ (x y z : α), dist x z ≤ dist x y + dist y z
• edist : α → α → ENNReal
• edist_dist : ∀ (x y : α), PseudoMetricSpace.edist x y = ENNReal.ofReal (dist x y)
• toUniformSpace : UniformSpace α
• uniformity_dist : uniformity α = ⨅ (ε : ℝ), ⨅ (_ : ε > 0), Filter.principal {p : α × α | dist p.1 p.2 < ε}
• toBornology : Bornology α
• cobounded_sets : (Bornology.cobounded α).sets = {s : Set α | ∃ (C : ℝ), ∀ x ∈ sᶜ, ∀ y ∈ sᶜ, dist x y ≤ C}

theorem PseudoMetricSpace.dist_self {α : Type u} [self : PseudoMetricSpace α] (x : α) : dist x x = 0

theorem PseudoMetricSpace.dist_comm {α : Type u} [self : PseudoMetricSpace α] (x : α) (y : α) : dist x y = dist y x

theorem PseudoMetricSpace.dist_triangle {α : Type u} [self : PseudoMetricSpace α] (x : α) (y : α) (z : α) : dist x z ≤ dist x y + dist y z

theorem PseudoMetricSpace.edist_dist {α : Type u} [self : PseudoMetricSpace α] (x : α) (y : α) : PseudoMetricSpace.edist x y = ENNReal.ofReal (dist x y)

theorem PseudoMetricSpace.uniformity_dist {α : Type u} [self : PseudoMetricSpace α] : uniformity α = ⨅ (ε : ℝ), ⨅ (_ : ε > 0), Filter.principal {p : α × α | dist p.1 p.2 < ε}

theorem PseudoMetricSpace.cobounded_sets {α : Type u} [self : PseudoMetricSpace α] : (Bornology.cobounded α).sets = {s : Set α | ∃ (C : ℝ), ∀ x ∈ sᶜ, ∀ y ∈ sᶜ, dist x y ≤ C}

theorem PseudoMetricSpace.ext {α : Type u_3} {m : PseudoMetricSpace α} {m' : PseudoMetricSpace α} (h : PseudoMetricSpace.toDist = PseudoMetricSpace.toDist) : m = m'

Two pseudo metric space structures with the same distance function coincide.

@[instance 200]

instance PseudoMetricSpace.toEDist {α : Type u} [PseudoMetricSpace α] : EDist α

Equations

• PseudoMetricSpace.toEDist = { edist := PseudoMetricSpace.edist }

def PseudoMetricSpace.ofDistTopology {α : Type u} [TopologicalSpace α] (dist : α → α → ℝ) (dist_self : ∀ (x : α), dist x x = 0) (dist_comm : ∀ (x y : α), dist x y = dist y x) (dist_triangle : ∀ (x y z : α), dist x z ≤ dist x y + dist y z) (H : ∀ (s : Set α), IsOpen s ↔ ∀ x ∈ s, ∃ ε > 0, ∀ (y : α), dist x y < ε → y ∈ s) : PseudoMetricSpace α

Construct a pseudo-metric space structure whose underlying topological space structure (definitionally) agrees which a pre-existing topology which is compatible with a given distance function.

Equations

• One or more equations did not get rendered due to their size.

@[simp]

theorem dist_self {α : Type u} [PseudoMetricSpace α] (x : α) : dist x x = 0

theorem dist_comm {α : Type u} [PseudoMetricSpace α] (x : α) (y : α) : dist x y = dist y x

theorem edist_dist {α : Type u} [PseudoMetricSpace α] (x : α) (y : α) : edist x y = ENNReal.ofReal (dist x y)

theorem dist_triangle {α : Type u} [PseudoMetricSpace α] (x : α) (y : α) (z : α) : dist x z ≤ dist x y + dist y z

theorem dist_triangle_left {α : Type u} [PseudoMetricSpace α] (x : α) (y : α) (z : α) : dist x y ≤ dist z x + dist z y

theorem dist_triangle_right {α : Type u} [PseudoMetricSpace α] (x : α) (y : α) (z : α) : dist x y ≤ dist x z + dist y z

theorem dist_triangle4 {α : Type u} [PseudoMetricSpace α] (x : α) (y : α) (z : α) (w : α) : dist x w ≤ dist x y + dist y z + dist z w

theorem dist_triangle4_left {α : Type u} [PseudoMetricSpace α] (x₁ : α) (y₁ : α) (x₂ : α) (y₂ : α) : dist x₂ y₂ ≤ dist x₁ y₁ + (dist x₁ x₂ + dist y₁ y₂)

theorem dist_triangle4_right {α : Type u} [PseudoMetricSpace α] (x₁ : α) (y₁ : α) (x₂ : α) (y₂ : α) : dist x₁ y₁ ≤ dist x₁ x₂ + dist y₁ y₂ + dist x₂ y₂

theorem dist_le_Ico_sum_dist {α : Type u} [PseudoMetricSpace α] (f : ℕ → α) {m : ℕ} {n : ℕ} (h : m ≤ n) : dist (f m) (f n) ≤ ∑ i ∈ Finset.Ico m n, dist (f i) (f (i + 1))

The triangle (polygon) inequality for sequences of points; Finset.Ico version.

theorem dist_le_range_sum_dist {α : Type u} [PseudoMetricSpace α] (f : ℕ → α) (n : ℕ) : dist (f 0) (f n) ≤ ∑ i ∈ Finset.range n, dist (f i) (f (i + 1))

The triangle (polygon) inequality for sequences of points; Finset.range version.

theorem dist_le_Ico_sum_of_dist_le {α : Type u} [PseudoMetricSpace α] {f : ℕ → α} {m : ℕ} {n : ℕ} (hmn : m ≤ n) {d : ℕ → ℝ} (hd : ∀ {k : ℕ}, m ≤ k → k < n → dist (f k) (f (k + 1)) ≤ d k) : dist (f m) (f n) ≤ ∑ i ∈ Finset.Ico m n, d i

A version of dist_le_Ico_sum_dist with each intermediate distance replaced with an upper estimate.

theorem dist_le_range_sum_of_dist_le {α : Type u} [PseudoMetricSpace α] {f : ℕ → α} (n : ℕ) {d : ℕ → ℝ} (hd : ∀ {k : ℕ}, k < n → dist (f k) (f (k + 1)) ≤ d k) : dist (f 0) (f n) ≤ ∑ i ∈ Finset.range n, d i

A version of dist_le_range_sum_dist with each intermediate distance replaced with an upper estimate.

theorem swap_dist {α : Type u} [PseudoMetricSpace α] : Function.swap dist = dist

theorem abs_dist_sub_le {α : Type u} [PseudoMetricSpace α] (x : α) (y : α) (z : α) : |dist x z - dist y z| ≤ dist x y

theorem dist_nonneg {α : Type u} [PseudoMetricSpace α] {x : α} {y : α} : 0 ≤ dist x y

def Mathlib.Meta.Positivity.evalDist : Mathlib.Meta.Positivity.PositivityExt

Extension for the positivity tactic: distances are nonnegative.

@[simp]

theorem abs_dist {α : Type u} [PseudoMetricSpace α] {a : α} {b : α} : |dist a b| = dist a b

class NNDist (α : Type u_3) : Type u_3

A version of Dist that takes value in ℝ≥0.

• nndist : α → α → NNReal

@[instance 100]

instance PseudoMetricSpace.toNNDist {α : Type u} [PseudoMetricSpace α] : NNDist α

Distance as a nonnegative real number.

Equations

• PseudoMetricSpace.toNNDist = { nndist := fun (a b : α) => ⟨dist a b, ⋯⟩ }

theorem dist_nndist {α : Type u} [PseudoMetricSpace α] (x : α) (y : α) : dist x y = ↑(nndist x y)

Express dist in terms of nndist

@[simp]

theorem coe_nndist {α : Type u} [PseudoMetricSpace α] (x : α) (y : α) : ↑(nndist x y) = dist x y

theorem edist_nndist {α : Type u} [PseudoMetricSpace α] (x : α) (y : α) : edist x y = ↑(nndist x y)

Express edist in terms of nndist

theorem nndist_edist {α : Type u} [PseudoMetricSpace α] (x : α) (y : α) : nndist x y = (edist x y).toNNReal

Express nndist in terms of edist

@[simp]

theorem coe_nnreal_ennreal_nndist {α : Type u} [PseudoMetricSpace α] (x : α) (y : α) : ↑(nndist x y) = edist x y

@[simp]

theorem edist_lt_coe {α : Type u} [PseudoMetricSpace α] {x : α} {y : α} {c : NNReal} : edist x y < ↑c ↔ nndist x y < c

@[simp]

theorem edist_le_coe {α : Type u} [PseudoMetricSpace α] {x : α} {y : α} {c : NNReal} : edist x y ≤ ↑c ↔ nndist x y ≤ c

theorem edist_lt_top {α : Type u_3} [PseudoMetricSpace α] (x : α) (y : α) : edist x y < ⊤

In a pseudometric space, the extended distance is always finite

theorem edist_ne_top {α : Type u} [PseudoMetricSpace α] (x : α) (y : α) : edist x y ≠ ⊤

In a pseudometric space, the extended distance is always finite

@[simp]

theorem nndist_self {α : Type u} [PseudoMetricSpace α] (a : α) : nndist a a = 0

nndist x x vanishes

@[simp]

theorem dist_lt_coe {α : Type u} [PseudoMetricSpace α] {x : α} {y : α} {c : NNReal} : dist x y < ↑c ↔ nndist x y < c

@[simp]

theorem dist_le_coe {α : Type u} [PseudoMetricSpace α] {x : α} {y : α} {c : NNReal} : dist x y ≤ ↑c ↔ nndist x y ≤ c

@[simp]

theorem edist_lt_ofReal {α : Type u} [PseudoMetricSpace α] {x : α} {y : α} {r : ℝ} : edist x y < ENNReal.ofReal r ↔ dist x y < r

@[simp]

theorem edist_le_ofReal {α : Type u} [PseudoMetricSpace α] {x : α} {y : α} {r : ℝ} (hr : 0 ≤ r) : edist x y ≤ ENNReal.ofReal r ↔ dist x y ≤ r

theorem nndist_dist {α : Type u} [PseudoMetricSpace α] (x : α) (y : α) : nndist x y = (dist x y).toNNReal

Express nndist in terms of dist

theorem nndist_comm {α : Type u} [PseudoMetricSpace α] (x : α) (y : α) : nndist x y = nndist y x

theorem nndist_triangle {α : Type u} [PseudoMetricSpace α] (x : α) (y : α) (z : α) : nndist x z ≤ nndist x y + nndist y z

Triangle inequality for the nonnegative distance

theorem nndist_triangle_left {α : Type u} [PseudoMetricSpace α] (x : α) (y : α) (z : α) : nndist x y ≤ nndist z x + nndist z y

theorem nndist_triangle_right {α : Type u} [PseudoMetricSpace α] (x : α) (y : α) (z : α) : nndist x y ≤ nndist x z + nndist y z

theorem dist_edist {α : Type u} [PseudoMetricSpace α] (x : α) (y : α) : dist x y = (edist x y).toReal

Express dist in terms of edist

def Metric.ball {α : Type u} [PseudoMetricSpace α] (x : α) (ε : ℝ) : Set α

ball x ε is the set of all points y with dist y x < ε

Equations

• Metric.ball x ε = {y : α | dist y x < ε}

@[simp]

theorem Metric.mem_ball {α : Type u} [PseudoMetricSpace α] {x : α} {y : α} {ε : ℝ} : y ∈ Metric.ball x ε ↔ dist y x < ε

theorem Metric.mem_ball' {α : Type u} [PseudoMetricSpace α] {x : α} {y : α} {ε : ℝ} : y ∈ Metric.ball x ε ↔ dist x y < ε

theorem Metric.pos_of_mem_ball {α : Type u} [PseudoMetricSpace α] {x : α} {y : α} {ε : ℝ} (hy : y ∈ Metric.ball x ε) : 0 < ε

theorem Metric.mem_ball_self {α : Type u} [PseudoMetricSpace α] {x : α} {ε : ℝ} (h : 0 < ε) : x ∈ Metric.ball x ε

@[simp]

theorem Metric.nonempty_ball {α : Type u} [PseudoMetricSpace α] {x : α} {ε : ℝ} : (Metric.ball x ε).Nonempty ↔ 0 < ε

@[simp]

theorem Metric.ball_eq_empty {α : Type u} [PseudoMetricSpace α] {x : α} {ε : ℝ} : Metric.ball x ε = ∅ ↔ ε ≤ 0

@[simp]

theorem Metric.ball_zero {α : Type u} [PseudoMetricSpace α] {x : α} : Metric.ball x 0 = ∅

theorem Metric.exists_lt_mem_ball_of_mem_ball {α : Type u} [PseudoMetricSpace α] {x : α} {y : α} {ε : ℝ} (h : x ∈ Metric.ball y ε) : ∃ ε' < ε, x ∈ Metric.ball y ε'

If a point belongs to an open ball, then there is a strictly smaller radius whose ball also contains it.

See also exists_lt_subset_ball.

theorem Metric.ball_eq_ball {α : Type u} [PseudoMetricSpace α] (ε : ℝ) (x : α) : UniformSpace.ball x {p : α × α | dist p.2 p.1 < ε} = Metric.ball x ε

theorem Metric.ball_eq_ball' {α : Type u} [PseudoMetricSpace α] (ε : ℝ) (x : α) : UniformSpace.ball x {p : α × α | dist p.1 p.2 < ε} = Metric.ball x ε

@[simp]

theorem Metric.iUnion_ball_nat {α : Type u} [PseudoMetricSpace α] (x : α) : ⋃ (n : ℕ), Metric.ball x ↑n = Set.univ

@[simp]

theorem Metric.iUnion_ball_nat_succ {α : Type u} [PseudoMetricSpace α] (x : α) : ⋃ (n : ℕ), Metric.ball x (↑n + 1) = Set.univ

def Metric.closedBall {α : Type u} [PseudoMetricSpace α] (x : α) (ε : ℝ) : Set α

closedBall x ε is the set of all points y with dist y x ≤ ε

Equations

• Metric.closedBall x ε = {y : α | dist y x ≤ ε}

@[simp]

theorem Metric.mem_closedBall {α : Type u} [PseudoMetricSpace α] {x : α} {y : α} {ε : ℝ} : y ∈ Metric.closedBall x ε ↔ dist y x ≤ ε

theorem Metric.mem_closedBall' {α : Type u} [PseudoMetricSpace α] {x : α} {y : α} {ε : ℝ} : y ∈ Metric.closedBall x ε ↔ dist x y ≤ ε

def Metric.sphere {α : Type u} [PseudoMetricSpace α] (x : α) (ε : ℝ) : Set α

sphere x ε is the set of all points y with dist y x = ε

Equations

• Metric.sphere x ε = {y : α | dist y x = ε}

@[simp]

theorem Metric.mem_sphere {α : Type u} [PseudoMetricSpace α] {x : α} {y : α} {ε : ℝ} : y ∈ Metric.sphere x ε ↔ dist y x = ε

theorem Metric.mem_sphere' {α : Type u} [PseudoMetricSpace α] {x : α} {y : α} {ε : ℝ} : y ∈ Metric.sphere x ε ↔ dist x y = ε

theorem Metric.ne_of_mem_sphere {α : Type u} [PseudoMetricSpace α] {x : α} {y : α} {ε : ℝ} (h : y ∈ Metric.sphere x ε) (hε : ε ≠ 0) : y ≠ x

theorem Metric.nonneg_of_mem_sphere {α : Type u} [PseudoMetricSpace α] {x : α} {y : α} {ε : ℝ} (hy : y ∈ Metric.sphere x ε) : 0 ≤ ε

@[simp]

theorem Metric.sphere_eq_empty_of_neg {α : Type u} [PseudoMetricSpace α] {x : α} {ε : ℝ} (hε : ε < 0) : Metric.sphere x ε = ∅

theorem Metric.sphere_eq_empty_of_subsingleton {α : Type u} [PseudoMetricSpace α] {x : α} {ε : ℝ} [Subsingleton α] (hε : ε ≠ 0) : Metric.sphere x ε = ∅

instance Metric.sphere_isEmpty_of_subsingleton {α : Type u} [PseudoMetricSpace α] {x : α} {ε : ℝ} [Subsingleton α] [NeZero ε] : IsEmpty ↑(Metric.sphere x ε)

Equations

• ⋯ = ⋯

theorem Metric.mem_closedBall_self {α : Type u} [PseudoMetricSpace α] {x : α} {ε : ℝ} (h : 0 ≤ ε) : x ∈ Metric.closedBall x ε

@[simp]

theorem Metric.nonempty_closedBall {α : Type u} [PseudoMetricSpace α] {x : α} {ε : ℝ} : (Metric.closedBall x ε).Nonempty ↔ 0 ≤ ε

@[simp]

theorem Metric.closedBall_eq_empty {α : Type u} [PseudoMetricSpace α] {x : α} {ε : ℝ} : Metric.closedBall x ε = ∅ ↔ ε < 0

theorem Metric.closedBall_eq_sphere_of_nonpos {α : Type u} [PseudoMetricSpace α] {x : α} {ε : ℝ} (hε : ε ≤ 0) : Metric.closedBall x ε = Metric.sphere x ε

Closed balls and spheres coincide when the radius is non-positive

theorem Metric.ball_subset_closedBall {α : Type u} [PseudoMetricSpace α] {x : α} {ε : ℝ} : Metric.ball x ε ⊆ Metric.closedBall x ε

theorem Metric.sphere_subset_closedBall {α : Type u} [PseudoMetricSpace α] {x : α} {ε : ℝ} : Metric.sphere x ε ⊆ Metric.closedBall x ε

theorem Metric.sphere_subset_ball {α : Type u} [PseudoMetricSpace α] {x : α} {r : ℝ} {R : ℝ} (h : r < R) : Metric.sphere x r ⊆ Metric.ball x R

theorem Metric.closedBall_disjoint_ball {α : Type u} [PseudoMetricSpace α] {x : α} {y : α} {δ : ℝ} {ε : ℝ} (h : δ + ε ≤ dist x y) : Disjoint (Metric.closedBall x δ) (Metric.ball y ε)

theorem Metric.ball_disjoint_closedBall {α : Type u} [PseudoMetricSpace α] {x : α} {y : α} {δ : ℝ} {ε : ℝ} (h : δ + ε ≤ dist x y) : Disjoint (Metric.ball x δ) (Metric.closedBall y ε)

theorem Metric.ball_disjoint_ball {α : Type u} [PseudoMetricSpace α] {x : α} {y : α} {δ : ℝ} {ε : ℝ} (h : δ + ε ≤ dist x y) : Disjoint (Metric.ball x δ) (Metric.ball y ε)

theorem Metric.closedBall_disjoint_closedBall {α : Type u} [PseudoMetricSpace α] {x : α} {y : α} {δ : ℝ} {ε : ℝ} (h : δ + ε < dist x y) : Disjoint (Metric.closedBall x δ) (Metric.closedBall y ε)

theorem Metric.sphere_disjoint_ball {α : Type u} [PseudoMetricSpace α] {x : α} {ε : ℝ} : Disjoint (Metric.sphere x ε) (Metric.ball x ε)

@[simp]

theorem Metric.ball_union_sphere {α : Type u} [PseudoMetricSpace α] {x : α} {ε : ℝ} : Metric.ball x ε ∪ Metric.sphere x ε = Metric.closedBall x ε

@[simp]

theorem Metric.sphere_union_ball {α : Type u} [PseudoMetricSpace α] {x : α} {ε : ℝ} : Metric.sphere x ε ∪ Metric.ball x ε = Metric.closedBall x ε

@[simp]

theorem Metric.closedBall_diff_sphere {α : Type u} [PseudoMetricSpace α] {x : α} {ε : ℝ} : Metric.closedBall x ε \ Metric.sphere x ε = Metric.ball x ε

@[simp]

theorem Metric.closedBall_diff_ball {α : Type u} [PseudoMetricSpace α] {x : α} {ε : ℝ} : Metric.closedBall x ε \ Metric.ball x ε = Metric.sphere x ε

theorem Metric.mem_ball_comm {α : Type u} [PseudoMetricSpace α] {x : α} {y : α} {ε : ℝ} : x ∈ Metric.ball y ε ↔ y ∈ Metric.ball x ε

theorem Metric.mem_closedBall_comm {α : Type u} [PseudoMetricSpace α] {x : α} {y : α} {ε : ℝ} : x ∈ Metric.closedBall y ε ↔ y ∈ Metric.closedBall x ε

theorem Metric.mem_sphere_comm {α : Type u} [PseudoMetricSpace α] {x : α} {y : α} {ε : ℝ} : x ∈ Metric.sphere y ε ↔ y ∈ Metric.sphere x ε

theorem Metric.ball_subset_ball {α : Type u} [PseudoMetricSpace α] {x : α} {ε₁ : ℝ} {ε₂ : ℝ} (h : ε₁ ≤ ε₂) : Metric.ball x ε₁ ⊆ Metric.ball x ε₂

theorem Metric.closedBall_eq_bInter_ball {α : Type u} [PseudoMetricSpace α] {x : α} {ε : ℝ} : Metric.closedBall x ε = ⋂ (δ : ℝ), ⋂ (_ : δ > ε), Metric.ball x δ

theorem Metric.ball_subset_ball' {α : Type u} [PseudoMetricSpace α] {x : α} {y : α} {ε₁ : ℝ} {ε₂ : ℝ} (h : ε₁ + dist x y ≤ ε₂) : Metric.ball x ε₁ ⊆ Metric.ball y ε₂

theorem Metric.closedBall_subset_closedBall {α : Type u} [PseudoMetricSpace α] {x : α} {ε₁ : ℝ} {ε₂ : ℝ} (h : ε₁ ≤ ε₂) : Metric.closedBall x ε₁ ⊆ Metric.closedBall x ε₂

theorem Metric.closedBall_subset_closedBall' {α : Type u} [PseudoMetricSpace α] {x : α} {y : α} {ε₁ : ℝ} {ε₂ : ℝ} (h : ε₁ + dist x y ≤ ε₂) : Metric.closedBall x ε₁ ⊆ Metric.closedBall y ε₂

theorem Metric.closedBall_subset_ball {α : Type u} [PseudoMetricSpace α] {x : α} {ε₁ : ℝ} {ε₂ : ℝ} (h : ε₁ < ε₂) : Metric.closedBall x ε₁ ⊆ Metric.ball x ε₂

theorem Metric.closedBall_subset_ball' {α : Type u} [PseudoMetricSpace α] {x : α} {y : α} {ε₁ : ℝ} {ε₂ : ℝ} (h : ε₁ + dist x y < ε₂) : Metric.closedBall x ε₁ ⊆ Metric.ball y ε₂

theorem Metric.dist_le_add_of_nonempty_closedBall_inter_closedBall {α : Type u} [PseudoMetricSpace α] {x : α} {y : α} {ε₁ : ℝ} {ε₂ : ℝ} (h : (Metric.closedBall x ε₁ ∩ Metric.closedBall y ε₂).Nonempty) : dist x y ≤ ε₁ + ε₂

theorem Metric.dist_lt_add_of_nonempty_closedBall_inter_ball {α : Type u} [PseudoMetricSpace α] {x : α} {y : α} {ε₁ : ℝ} {ε₂ : ℝ} (h : (Metric.closedBall x ε₁ ∩ Metric.ball y ε₂).Nonempty) : dist x y < ε₁ + ε₂

theorem Metric.dist_lt_add_of_nonempty_ball_inter_closedBall {α : Type u} [PseudoMetricSpace α] {x : α} {y : α} {ε₁ : ℝ} {ε₂ : ℝ} (h : (Metric.ball x ε₁ ∩ Metric.closedBall y ε₂).Nonempty) : dist x y < ε₁ + ε₂

theorem Metric.dist_lt_add_of_nonempty_ball_inter_ball {α : Type u} [PseudoMetricSpace α] {x : α} {y : α} {ε₁ : ℝ} {ε₂ : ℝ} (h : (Metric.ball x ε₁ ∩ Metric.ball y ε₂).Nonempty) : dist x y < ε₁ + ε₂

@[simp]

theorem Metric.iUnion_closedBall_nat {α : Type u} [PseudoMetricSpace α] (x : α) : ⋃ (n : ℕ), Metric.closedBall x ↑n = Set.univ

theorem Metric.iUnion_inter_closedBall_nat {α : Type u} [PseudoMetricSpace α] (s : Set α) (x : α) : ⋃ (n : ℕ), s ∩ Metric.closedBall x ↑n = s

theorem Metric.ball_subset {α : Type u} [PseudoMetricSpace α] {x : α} {y : α} {ε₁ : ℝ} {ε₂ : ℝ} (h : dist x y ≤ ε₂ - ε₁) : Metric.ball x ε₁ ⊆ Metric.ball y ε₂

theorem Metric.ball_half_subset {α : Type u} [PseudoMetricSpace α] {x : α} {ε : ℝ} (y : α) (h : y ∈ Metric.ball x (ε / 2)) : Metric.ball y (ε / 2) ⊆ Metric.ball x ε

theorem Metric.exists_ball_subset_ball {α : Type u} [PseudoMetricSpace α] {x : α} {y : α} {ε : ℝ} (h : y ∈ Metric.ball x ε) : ∃ ε' > 0, Metric.ball y ε' ⊆ Metric.ball x ε

theorem Metric.forall_of_forall_mem_closedBall {α : Type u} [PseudoMetricSpace α] (p : α → Prop) (x : α) (H : ∃ᶠ (R : ℝ) in Filter.atTop, ∀ y ∈ Metric.closedBall x R, p y) (y : α) : p y

If a property holds for all points in closed balls of arbitrarily large radii, then it holds for all points.

theorem Metric.forall_of_forall_mem_ball {α : Type u} [PseudoMetricSpace α] (p : α → Prop) (x : α) (H : ∃ᶠ (R : ℝ) in Filter.atTop, ∀ y ∈ Metric.ball x R, p y) (y : α) : p y

If a property holds for all points in balls of arbitrarily large radii, then it holds for all points.

theorem Metric.isBounded_iff {α : Type u} [PseudoMetricSpace α] {s : Set α} : Bornology.IsBounded s ↔ ∃ (C : ℝ), ∀ ⦃x : α⦄, x ∈ s → ∀ ⦃y : α⦄, y ∈ s → dist x y ≤ C

theorem Metric.isBounded_iff_eventually {α : Type u} [PseudoMetricSpace α] {s : Set α} : Bornology.IsBounded s ↔ ∀ᶠ (C : ℝ) in Filter.atTop, ∀ ⦃x : α⦄, x ∈ s → ∀ ⦃y : α⦄, y ∈ s → dist x y ≤ C

theorem Metric.isBounded_iff_exists_ge {α : Type u} [PseudoMetricSpace α] {s : Set α} (c : ℝ) : Bornology.IsBounded s ↔ ∃ (C : ℝ), c ≤ C ∧ ∀ ⦃x : α⦄, x ∈ s → ∀ ⦃y : α⦄, y ∈ s → dist x y ≤ C

theorem Metric.isBounded_iff_nndist {α : Type u} [PseudoMetricSpace α] {s : Set α} : Bornology.IsBounded s ↔ ∃ (C : NNReal), ∀ ⦃x : α⦄, x ∈ s → ∀ ⦃y : α⦄, y ∈ s → nndist x y ≤ C

theorem Metric.toUniformSpace_eq {α : Type u} [PseudoMetricSpace α] : PseudoMetricSpace.toUniformSpace = UniformSpace.ofDist dist ⋯ ⋯ ⋯

theorem Metric.uniformity_basis_dist {α : Type u} [PseudoMetricSpace α] : (uniformity α).HasBasis (fun (ε : ℝ) => 0 < ε) fun (ε : ℝ) => {p : α × α | dist p.1 p.2 < ε}

theorem Metric.mk_uniformity_basis {α : Type u} [PseudoMetricSpace α] {β : Type u_3} {p : β → Prop} {f : β → ℝ} (hf₀ : ∀ (i : β), p i → 0 < f i) (hf : ∀ ⦃ε : ℝ⦄, 0 < ε → ∃ (i : β), p i ∧ f i ≤ ε) : (uniformity α).HasBasis p fun (i : β) => {p : α × α | dist p.1 p.2 < f i}

Given f : β → ℝ, if f sends {i | p i} to a set of positive numbers accumulating to zero, then f i-neighborhoods of the diagonal form a basis of 𝓤 α.

For specific bases see uniformity_basis_dist, uniformity_basis_dist_inv_nat_succ, and uniformity_basis_dist_inv_nat_pos.

theorem Metric.uniformity_basis_dist_rat {α : Type u} [PseudoMetricSpace α] : (uniformity α).HasBasis (fun (r : ℚ) => 0 < r) fun (r : ℚ) => {p : α × α | dist p.1 p.2 < ↑r}

theorem Metric.uniformity_basis_dist_inv_nat_succ {α : Type u} [PseudoMetricSpace α] : (uniformity α).HasBasis (fun (x : ℕ) => True) fun (n : ℕ) => {p : α × α | dist p.1 p.2 < 1 / (↑n + 1)}

theorem Metric.uniformity_basis_dist_inv_nat_pos {α : Type u} [PseudoMetricSpace α] : (uniformity α).HasBasis (fun (n : ℕ) => 0 < n) fun (n : ℕ) => {p : α × α | dist p.1 p.2 < 1 / ↑n}

theorem Metric.uniformity_basis_dist_pow {α : Type u} [PseudoMetricSpace α] {r : ℝ} (h0 : 0 < r) (h1 : r < 1) : (uniformity α).HasBasis (fun (x : ℕ) => True) fun (n : ℕ) => {p : α × α | dist p.1 p.2 < r ^ n}

theorem Metric.uniformity_basis_dist_lt {α : Type u} [PseudoMetricSpace α] {R : ℝ} (hR : 0 < R) : (uniformity α).HasBasis (fun (r : ℝ) => 0 < r ∧ r < R) fun (r : ℝ) => {p : α × α | dist p.1 p.2 < r}

theorem Metric.mk_uniformity_basis_le {α : Type u} [PseudoMetricSpace α] {β : Type u_3} {p : β → Prop} {f : β → ℝ} (hf₀ : ∀ (x : β), p x → 0 < f x) (hf : ∀ (ε : ℝ), 0 < ε → ∃ (x : β), p x ∧ f x ≤ ε) : (uniformity α).HasBasis p fun (x : β) => {p : α × α | dist p.1 p.2 ≤ f x}

Given f : β → ℝ, if f sends {i | p i} to a set of positive numbers accumulating to zero, then closed neighborhoods of the diagonal of sizes {f i | p i} form a basis of 𝓤 α.

Currently we have only one specific basis uniformity_basis_dist_le based on this constructor. More can be easily added if needed in the future.

theorem Metric.uniformity_basis_dist_le {α : Type u} [PseudoMetricSpace α] : (uniformity α).HasBasis (fun (x : ℝ) => 0 < x) fun (ε : ℝ) => {p : α × α | dist p.1 p.2 ≤ ε}

Constant size closed neighborhoods of the diagonal form a basis of the uniformity filter.

theorem Metric.uniformity_basis_dist_le_pow {α : Type u} [PseudoMetricSpace α] {r : ℝ} (h0 : 0 < r) (h1 : r < 1) : (uniformity α).HasBasis (fun (x : ℕ) => True) fun (n : ℕ) => {p : α × α | dist p.1 p.2 ≤ r ^ n}

theorem Metric.mem_uniformity_dist {α : Type u} [PseudoMetricSpace α] {s : Set (α × α)} : s ∈ uniformity α ↔ ∃ ε > 0, ∀ {a b : α}, dist a b < ε → (a, b) ∈ s

theorem Metric.dist_mem_uniformity {α : Type u} [PseudoMetricSpace α] {ε : ℝ} (ε0 : 0 < ε) : {p : α × α | dist p.1 p.2 < ε} ∈ uniformity α

A constant size neighborhood of the diagonal is an entourage.

theorem Metric.uniformContinuous_iff {α : Type u} {β : Type v} [PseudoMetricSpace α] [PseudoMetricSpace β] {f : α → β} : UniformContinuous f ↔ ∀ ε > 0, ∃ δ > 0, ∀ {a b : α}, dist a b < δ → dist (f a) (f b) < ε

theorem Metric.uniformContinuousOn_iff {α : Type u} {β : Type v} [PseudoMetricSpace α] [PseudoMetricSpace β] {f : α → β} {s : Set α} : UniformContinuousOn f s ↔ ∀ ε > 0, ∃ δ > 0, ∀ x ∈ s, ∀ y ∈ s, dist x y < δ → dist (f x) (f y) < ε

theorem Metric.uniformContinuousOn_iff_le {α : Type u} {β : Type v} [PseudoMetricSpace α] [PseudoMetricSpace β] {f : α → β} {s : Set α} : UniformContinuousOn f s ↔ ∀ ε > 0, ∃ δ > 0, ∀ x ∈ s, ∀ y ∈ s, dist x y ≤ δ → dist (f x) (f y) ≤ ε

theorem Metric.uniformInducing_iff {α : Type u} {β : Type v} [PseudoMetricSpace α] [PseudoMetricSpace β] {f : α → β} : UniformInducing f ↔ UniformContinuous f ∧ ∀ δ > 0, ∃ ε > 0, ∀ {a b : α}, dist (f a) (f b) < ε → dist a b < δ

theorem Metric.uniformEmbedding_iff {α : Type u} {β : Type v} [PseudoMetricSpace α] [PseudoMetricSpace β] {f : α → β} : UniformEmbedding f ↔ Function.Injective f ∧ UniformContinuous f ∧ ∀ δ > 0, ∃ ε > 0, ∀ {a b : α}, dist (f a) (f b) < ε → dist a b < δ

theorem Metric.controlled_of_uniformEmbedding {α : Type u} {β : Type v} [PseudoMetricSpace α] [PseudoMetricSpace β] {f : α → β} (h : UniformEmbedding f) : (∀ ε > 0, ∃ δ > 0, ∀ {a b : α}, dist a b < δ → dist (f a) (f b) < ε) ∧ ∀ δ > 0, ∃ ε > 0, ∀ {a b : α}, dist (f a) (f b) < ε → dist a b < δ

If a map between pseudometric spaces is a uniform embedding then the distance between f x and f y is controlled in terms of the distance between x and y.

theorem Metric.totallyBounded_iff {α : Type u} [PseudoMetricSpace α] {s : Set α} : TotallyBounded s ↔ ∀ ε > 0, ∃ (t : Set α), t.Finite ∧ s ⊆ ⋃ y ∈ t, Metric.ball y ε

theorem Metric.totallyBounded_of_finite_discretization {α : Type u} [PseudoMetricSpace α] {s : Set α} (H : ∀ ε > 0, ∃ (β : Type u) (x : Fintype β) (F : ↑s → β), ∀ (x y : ↑s), F x = F y → dist ↑x ↑y < ε) : TotallyBounded s

A pseudometric space is totally bounded if one can reconstruct up to any ε>0 any element of the space from finitely many data.

theorem Metric.finite_approx_of_totallyBounded {α : Type u} [PseudoMetricSpace α] {s : Set α} (hs : TotallyBounded s) (ε : ℝ) : ε > 0 → ∃ t ⊆ s, t.Finite ∧ s ⊆ ⋃ y ∈ t, Metric.ball y ε

theorem Metric.tendstoUniformlyOnFilter_iff {α : Type u} {β : Type v} {ι : Type u_2} [PseudoMetricSpace α] {F : ι → β → α} {f : β → α} {p : Filter ι} {p' : Filter β} : TendstoUniformlyOnFilter F f p p' ↔ ∀ ε > 0, ∀ᶠ (n : ι × β) in p ×ˢ p', dist (f n.2) (F n.1 n.2) < ε

Expressing uniform convergence using dist

theorem Metric.tendstoLocallyUniformlyOn_iff {α : Type u} {β : Type v} {ι : Type u_2} [PseudoMetricSpace α] [TopologicalSpace β] {F : ι → β → α} {f : β → α} {p : Filter ι} {s : Set β} : TendstoLocallyUniformlyOn F f p s ↔ ∀ ε > 0, ∀ x ∈ s, ∃ t ∈ nhdsWithin x s, ∀ᶠ (n : ι) in p, ∀ y ∈ t, dist (f y) (F n y) < ε

Expressing locally uniform convergence on a set using dist.

theorem Metric.tendstoUniformlyOn_iff {α : Type u} {β : Type v} {ι : Type u_2} [PseudoMetricSpace α] {F : ι → β → α} {f : β → α} {p : Filter ι} {s : Set β} : TendstoUniformlyOn F f p s ↔ ∀ ε > 0, ∀ᶠ (n : ι) in p, ∀ x ∈ s, dist (f x) (F n x) < ε

Expressing uniform convergence on a set using dist.

theorem Metric.tendstoLocallyUniformly_iff {α : Type u} {β : Type v} {ι : Type u_2} [PseudoMetricSpace α] [TopologicalSpace β] {F : ι → β → α} {f : β → α} {p : Filter ι} : TendstoLocallyUniformly F f p ↔ ∀ ε > 0, ∀ (x : β), ∃ t ∈ nhds x, ∀ᶠ (n : ι) in p, ∀ y ∈ t, dist (f y) (F n y) < ε

Expressing locally uniform convergence using dist.

theorem Metric.tendstoUniformly_iff {α : Type u} {β : Type v} {ι : Type u_2} [PseudoMetricSpace α] {F : ι → β → α} {f : β → α} {p : Filter ι} : TendstoUniformly F f p ↔ ∀ ε > 0, ∀ᶠ (n : ι) in p, ∀ (x : β), dist (f x) (F n x) < ε

Expressing uniform convergence using dist.

theorem Metric.cauchy_iff {α : Type u} [PseudoMetricSpace α] {f : Filter α} : Cauchy f ↔ f.NeBot ∧ ∀ ε > 0, ∃ t ∈ f, ∀ x ∈ t, ∀ y ∈ t, dist x y < ε

theorem Metric.nhds_basis_ball {α : Type u} [PseudoMetricSpace α] {x : α} : (nhds x).HasBasis (fun (x : ℝ) => 0 < x) (Metric.ball x)

theorem Metric.mem_nhds_iff {α : Type u} [PseudoMetricSpace α] {x : α} {s : Set α} : s ∈ nhds x ↔ ∃ ε > 0, Metric.ball x ε ⊆ s

theorem Metric.eventually_nhds_iff {α : Type u} [PseudoMetricSpace α] {x : α} {p : α → Prop} : (∀ᶠ (y : α) in nhds x, p y) ↔ ∃ ε > 0, ∀ ⦃y : α⦄, dist y x < ε → p y

theorem Metric.eventually_nhds_iff_ball {α : Type u} [PseudoMetricSpace α] {x : α} {p : α → Prop} : (∀ᶠ (y : α) in nhds x, p y) ↔ ∃ ε > 0, ∀ y ∈ Metric.ball x ε, p y

theorem Metric.eventually_nhds_prod_iff {α : Type u} {ι : Type u_2} [PseudoMetricSpace α] {f : Filter ι} {x₀ : α} {p : α × ι → Prop} : (∀ᶠ (x : α × ι) in nhds x₀ ×ˢ f, p x) ↔ ∃ ε > 0, ∃ (pa : ι → Prop), (∀ᶠ (i : ι) in f, pa i) ∧ ∀ {x : α}, dist x x₀ < ε → ∀ {i : ι}, pa i → p (x, i)

A version of Filter.eventually_prod_iff where the first filter consists of neighborhoods in a pseudo-metric space.

theorem Metric.eventually_prod_nhds_iff {α : Type u} {ι : Type u_2} [PseudoMetricSpace α] {f : Filter ι} {x₀ : α} {p : ι × α → Prop} : (∀ᶠ (x : ι × α) in f ×ˢ nhds x₀, p x) ↔ ∃ (pa : ι → Prop), (∀ᶠ (i : ι) in f, pa i) ∧ ∃ ε > 0, ∀ {i : ι}, pa i → ∀ {x : α}, dist x x₀ < ε → p (i, x)

A version of Filter.eventually_prod_iff where the second filter consists of neighborhoods in a pseudo-metric space.

theorem Metric.nhds_basis_closedBall {α : Type u} [PseudoMetricSpace α] {x : α} : (nhds x).HasBasis (fun (ε : ℝ) => 0 < ε) (Metric.closedBall x)

theorem Metric.nhds_basis_ball_inv_nat_succ {α : Type u} [PseudoMetricSpace α] {x : α} : (nhds x).HasBasis (fun (x : ℕ) => True) fun (n : ℕ) => Metric.ball x (1 / (↑n + 1))

theorem Metric.nhds_basis_ball_inv_nat_pos {α : Type u} [PseudoMetricSpace α] {x : α} : (nhds x).HasBasis (fun (n : ℕ) => 0 < n) fun (n : ℕ) => Metric.ball x (1 / ↑n)

theorem Metric.nhds_basis_ball_pow {α : Type u} [PseudoMetricSpace α] {x : α} {r : ℝ} (h0 : 0 < r) (h1 : r < 1) : (nhds x).HasBasis (fun (x : ℕ) => True) fun (n : ℕ) => Metric.ball x (r ^ n)

theorem Metric.nhds_basis_closedBall_pow {α : Type u} [PseudoMetricSpace α] {x : α} {r : ℝ} (h0 : 0 < r) (h1 : r < 1) : (nhds x).HasBasis (fun (x : ℕ) => True) fun (n : ℕ) => Metric.closedBall x (r ^ n)

theorem Metric.isOpen_iff {α : Type u} [PseudoMetricSpace α] {s : Set α} : IsOpen s ↔ ∀ x ∈ s, ∃ ε > 0, Metric.ball x ε ⊆ s

theorem Metric.isOpen_ball {α : Type u} [PseudoMetricSpace α] {x : α} {ε : ℝ} : IsOpen (Metric.ball x ε)

theorem Metric.ball_mem_nhds {α : Type u} [PseudoMetricSpace α] (x : α) {ε : ℝ} (ε0 : 0 < ε) : Metric.ball x ε ∈ nhds x

theorem Metric.closedBall_mem_nhds {α : Type u} [PseudoMetricSpace α] (x : α) {ε : ℝ} (ε0 : 0 < ε) : Metric.closedBall x ε ∈ nhds x

theorem Metric.closedBall_mem_nhds_of_mem {α : Type u} [PseudoMetricSpace α] {x : α} {c : α} {ε : ℝ} (h : x ∈ Metric.ball c ε) : Metric.closedBall c ε ∈ nhds x

theorem Metric.nhdsWithin_basis_ball {α : Type u} [PseudoMetricSpace α] {x : α} {s : Set α} : (nhdsWithin x s).HasBasis (fun (ε : ℝ) => 0 < ε) fun (ε : ℝ) => Metric.ball x ε ∩ s

theorem Metric.mem_nhdsWithin_iff {α : Type u} [PseudoMetricSpace α] {x : α} {s : Set α} {t : Set α} : s ∈ nhdsWithin x t ↔ ∃ ε > 0, Metric.ball x ε ∩ t ⊆ s

theorem Metric.tendsto_nhdsWithin_nhdsWithin {α : Type u} {β : Type v} [PseudoMetricSpace α] {s : Set α} [PseudoMetricSpace β] {t : Set β} {f : α → β} {a : α} {b : β} : Filter.Tendsto f (nhdsWithin a s) (nhdsWithin b t) ↔ ∀ ε > 0, ∃ δ > 0, ∀ {x : α}, x ∈ s → dist x a < δ → f x ∈ t ∧ dist (f x) b < ε

theorem Metric.tendsto_nhdsWithin_nhds {α : Type u} {β : Type v} [PseudoMetricSpace α] {s : Set α} [PseudoMetricSpace β] {f : α → β} {a : α} {b : β} : Filter.Tendsto f (nhdsWithin a s) (nhds b) ↔ ∀ ε > 0, ∃ δ > 0, ∀ {x : α}, x ∈ s → dist x a < δ → dist (f x) b < ε

theorem Metric.tendsto_nhds_nhds {α : Type u} {β : Type v} [PseudoMetricSpace α] [PseudoMetricSpace β] {f : α → β} {a : α} {b : β} : Filter.Tendsto f (nhds a) (nhds b) ↔ ∀ ε > 0, ∃ δ > 0, ∀ {x : α}, dist x a < δ → dist (f x) b < ε

theorem Metric.continuousAt_iff {α : Type u} {β : Type v} [PseudoMetricSpace α] [PseudoMetricSpace β] {f : α → β} {a : α} : ContinuousAt f a ↔ ∀ ε > 0, ∃ δ > 0, ∀ {x : α}, dist x a < δ → dist (f x) (f a) < ε

theorem Metric.continuousWithinAt_iff {α : Type u} {β : Type v} [PseudoMetricSpace α] [PseudoMetricSpace β] {f : α → β} {a : α} {s : Set α} : ContinuousWithinAt f s a ↔ ∀ ε > 0, ∃ δ > 0, ∀ {x : α}, x ∈ s → dist x a < δ → dist (f x) (f a) < ε

theorem Metric.continuousOn_iff {α : Type u} {β : Type v} [PseudoMetricSpace α] [PseudoMetricSpace β] {f : α → β} {s : Set α} : ContinuousOn f s ↔ ∀ b ∈ s, ∀ ε > 0, ∃ δ > 0, ∀ a ∈ s, dist a b < δ → dist (f a) (f b) < ε

theorem Metric.continuous_iff {α : Type u} {β : Type v} [PseudoMetricSpace α] [PseudoMetricSpace β] {f : α → β} : Continuous f ↔ ∀ (b : α), ∀ ε > 0, ∃ δ > 0, ∀ (a : α), dist a b < δ → dist (f a) (f b) < ε

theorem Metric.tendsto_nhds {α : Type u} {β : Type v} [PseudoMetricSpace α] {f : Filter β} {u : β → α} {a : α} : Filter.Tendsto u f (nhds a) ↔ ∀ ε > 0, ∀ᶠ (x : β) in f, dist (u x) a < ε

theorem Metric.continuousAt_iff' {α : Type u} {β : Type v} [PseudoMetricSpace α] [TopologicalSpace β] {f : β → α} {b : β} : ContinuousAt f b ↔ ∀ ε > 0, ∀ᶠ (x : β) in nhds b, dist (f x) (f b) < ε

theorem Metric.continuousWithinAt_iff' {α : Type u} {β : Type v} [PseudoMetricSpace α] [TopologicalSpace β] {f : β → α} {b : β} {s : Set β} : ContinuousWithinAt f s b ↔ ∀ ε > 0, ∀ᶠ (x : β) in nhdsWithin b s, dist (f x) (f b) < ε

theorem Metric.continuousOn_iff' {α : Type u} {β : Type v} [PseudoMetricSpace α] [TopologicalSpace β] {f : β → α} {s : Set β} : ContinuousOn f s ↔ ∀ b ∈ s, ∀ ε > 0, ∀ᶠ (x : β) in nhdsWithin b s, dist (f x) (f b) < ε

theorem Metric.continuous_iff' {α : Type u} {β : Type v} [PseudoMetricSpace α] [TopologicalSpace β] {f : β → α} : Continuous f ↔ ∀ (a : β), ∀ ε > 0, ∀ᶠ (x : β) in nhds a, dist (f x) (f a) < ε

theorem Metric.tendsto_atTop {α : Type u} {β : Type v} [PseudoMetricSpace α] [Nonempty β] [SemilatticeSup β] {u : β → α} {a : α} : Filter.Tendsto u Filter.atTop (nhds a) ↔ ∀ ε > 0, ∃ (N : β), ∀ n ≥ N, dist (u n) a < ε

theorem Metric.tendsto_atTop' {α : Type u} {β : Type v} [PseudoMetricSpace α] [Nonempty β] [SemilatticeSup β] [NoMaxOrder β] {u : β → α} {a : α} : Filter.Tendsto u Filter.atTop (nhds a) ↔ ∀ ε > 0, ∃ (N : β), ∀ n > N, dist (u n) a < ε

A variant of tendsto_atTop that uses ∃ N, ∀ n > N, ... rather than ∃ N, ∀ n ≥ N, ...

theorem Metric.isOpen_singleton_iff {α : Type u_3} [PseudoMetricSpace α] {x : α} : IsOpen {x} ↔ ∃ ε > 0, ∀ (y : α), dist y x < ε → y = x

theorem Metric.exists_ball_inter_eq_singleton_of_mem_discrete {α : Type u} [PseudoMetricSpace α] {s : Set α} [DiscreteTopology ↑s] {x : α} (hx : x ∈ s) : ∃ ε > 0, Metric.ball x ε ∩ s = {x}

Given a point x in a discrete subset s of a pseudometric space, there is an open ball centered at x and intersecting s only at x.

theorem Metric.exists_closedBall_inter_eq_singleton_of_discrete {α : Type u} [PseudoMetricSpace α] {s : Set α} [DiscreteTopology ↑s] {x : α} (hx : x ∈ s) : ∃ ε > 0, Metric.closedBall x ε ∩ s = {x}

Given a point x in a discrete subset s of a pseudometric space, there is a closed ball of positive radius centered at x and intersecting s only at x.

theorem Dense.exists_dist_lt {α : Type u} [PseudoMetricSpace α] {s : Set α} (hs : Dense s) (x : α) {ε : ℝ} (hε : 0 < ε) : ∃ y ∈ s, dist x y < ε

theorem DenseRange.exists_dist_lt {α : Type u} [PseudoMetricSpace α] {β : Type u_3} {f : β → α} (hf : DenseRange f) (x : α) {ε : ℝ} (hε : 0 < ε) : ∃ (y : β), dist x (f y) < ε

theorem Metric.uniformSpace_eq_bot {α : Type u} [PseudoMetricSpace α] : PseudoMetricSpace.toUniformSpace = ⊥ ↔ ∃ (r : ℝ), 0 < r ∧ Pairwise fun (x x_1 : α) => r ≤ dist x x_1

(Pseudo) metric space has discrete UniformSpace structure iff the distances between distinct points are uniformly bounded away from zero.

theorem DiscreteTopology.of_forall_le_dist {α : Type u_3} [PseudoMetricSpace α] {r : ℝ} (hpos : 0 < r) (hr : Pairwise fun (x x_1 : α) => r ≤ dist x x_1) : DiscreteTopology α

If the distances between distinct points in a (pseudo) metric space are uniformly bounded away from zero, then the space has discrete topology.

theorem Metric.uniformity_edist_aux {α : Type u_3} (d : α → α → NNReal) : ⨅ (ε : ℝ), ⨅ (_ : ε > 0), Filter.principal {p : α × α | ↑(d p.1 p.2) < ε} = ⨅ (ε : ENNReal), ⨅ (_ : ε > 0), Filter.principal {p : α × α | ↑(d p.1 p.2) < ε}

theorem Metric.uniformity_edist {α : Type u} [PseudoMetricSpace α] : uniformity α = ⨅ (ε : ENNReal), ⨅ (_ : ε > 0), Filter.principal {p : α × α | edist p.1 p.2 < ε}

@[instance 100]

instance PseudoMetricSpace.toPseudoEMetricSpace {α : Type u} [PseudoMetricSpace α] : PseudoEMetricSpace α

A pseudometric space induces a pseudoemetric space

Equations

• PseudoMetricSpace.toPseudoEMetricSpace = let __src := inst; PseudoEMetricSpace.mk ⋯ ⋯ ⋯ PseudoMetricSpace.toUniformSpace ⋯

theorem Metric.eball_top_eq_univ {α : Type u} [PseudoMetricSpace α] (x : α) : EMetric.ball x ⊤ = Set.univ

In a pseudometric space, an open ball of infinite radius is the whole space

@[simp]

theorem Metric.emetric_ball {α : Type u} [PseudoMetricSpace α] {x : α} {ε : ℝ} : EMetric.ball x (ENNReal.ofReal ε) = Metric.ball x ε

Balls defined using the distance or the edistance coincide

@[simp]

theorem Metric.emetric_ball_nnreal {α : Type u} [PseudoMetricSpace α] {x : α} {ε : NNReal} : EMetric.ball x ↑ε = Metric.ball x ↑ε

Balls defined using the distance or the edistance coincide

theorem Metric.emetric_closedBall {α : Type u} [PseudoMetricSpace α] {x : α} {ε : ℝ} (h : 0 ≤ ε) : EMetric.closedBall x (ENNReal.ofReal ε) = Metric.closedBall x ε

Closed balls defined using the distance or the edistance coincide

@[simp]

theorem Metric.emetric_closedBall_nnreal {α : Type u} [PseudoMetricSpace α] {x : α} {ε : NNReal} : EMetric.closedBall x ↑ε = Metric.closedBall x ↑ε

Closed balls defined using the distance or the edistance coincide

@[simp]

theorem Metric.emetric_ball_top {α : Type u} [PseudoMetricSpace α] (x : α) : EMetric.ball x ⊤ = Set.univ

theorem Metric.inseparable_iff {α : Type u} [PseudoMetricSpace α] {x : α} {y : α} : Inseparable x y ↔ dist x y = 0

@[reducible, inline]

abbrev PseudoMetricSpace.replaceUniformity {α : Type u_3} [U : UniformSpace α] (m : PseudoMetricSpace α) (H : uniformity α = uniformity α) : PseudoMetricSpace α

Build a new pseudometric space from an old one where the bundled uniform structure is provably (but typically non-definitionaly) equal to some given uniform structure. See Note [forgetful inheritance].

Equations

• m.replaceUniformity H = PseudoMetricSpace.mk ⋯ ⋯ ⋯ PseudoMetricSpace.edist ⋯ U ⋯ PseudoMetricSpace.toBornology ⋯

theorem PseudoMetricSpace.replaceUniformity_eq {α : Type u_3} [U : UniformSpace α] (m : PseudoMetricSpace α) (H : uniformity α = uniformity α) : m.replaceUniformity H = m

@[reducible, inline]

abbrev PseudoMetricSpace.replaceTopology {γ : Type u_3} [U : TopologicalSpace γ] (m : PseudoMetricSpace γ) (H : U = UniformSpace.toTopologicalSpace) : PseudoMetricSpace γ

Build a new pseudo metric space from an old one where the bundled topological structure is provably (but typically non-definitionaly) equal to some given topological structure. See Note [forgetful inheritance].

Equations

• m.replaceTopology H = m.replaceUniformity ⋯

theorem PseudoMetricSpace.replaceTopology_eq {γ : Type u_3} [U : TopologicalSpace γ] (m : PseudoMetricSpace γ) (H : U = UniformSpace.toTopologicalSpace) : m.replaceTopology H = m

@[reducible, inline]

abbrev PseudoEMetricSpace.toPseudoMetricSpaceOfDist {α : Type u} [e : PseudoEMetricSpace α] (dist : α → α → ℝ) (edist_ne_top : ∀ (x y : α), edist x y ≠ ⊤) (h : ∀ (x y : α), dist x y = (edist x y).toReal) : PseudoMetricSpace α

One gets a pseudometric space from an emetric space if the edistance is everywhere finite, by pushing the edistance to reals. We set it up so that the edist and the uniformity are defeq in the pseudometric space and the pseudoemetric space. In this definition, the distance is given separately, to be able to prescribe some expression which is not defeq to the push-forward of the edistance to reals. See note [reducible non-instances].

Equations

• PseudoEMetricSpace.toPseudoMetricSpaceOfDist dist edist_ne_top h = PseudoMetricSpace.mk ⋯ ⋯ ⋯ edist ⋯ PseudoEMetricSpace.toUniformSpace ⋯ (Bornology.ofDist dist ⋯ ⋯) ⋯

@[reducible, inline]

abbrev PseudoEMetricSpace.toPseudoMetricSpace {α : Type u} [PseudoEMetricSpace α] (h : ∀ (x y : α), edist x y ≠ ⊤) : PseudoMetricSpace α

One gets a pseudometric space from an emetric space if the edistance is everywhere finite, by pushing the edistance to reals. We set it up so that the edist and the uniformity are defeq in the pseudometric space and the emetric space.

Equations

• PseudoEMetricSpace.toPseudoMetricSpace h = PseudoEMetricSpace.toPseudoMetricSpaceOfDist (fun (x y : α) => (edist x y).toReal) h ⋯

@[reducible, inline]

abbrev PseudoMetricSpace.replaceBornology {α : Type u_3} [B : Bornology α] (m : PseudoMetricSpace α) (H : ∀ (s : Set α), Bornology.IsBounded s ↔ Bornology.IsBounded s) : PseudoMetricSpace α

Build a new pseudometric space from an old one where the bundled bornology structure is provably (but typically non-definitionaly) equal to some given bornology structure. See Note [forgetful inheritance].

Equations

• m.replaceBornology H = PseudoMetricSpace.mk ⋯ ⋯ ⋯ PseudoMetricSpace.edist ⋯ PseudoMetricSpace.toUniformSpace ⋯ B ⋯

theorem PseudoMetricSpace.replaceBornology_eq {α : Type u_3} [m : PseudoMetricSpace α] [B : Bornology α] (H : ∀ (s : Set α), Bornology.IsBounded s ↔ Bornology.IsBounded s) : m.replaceBornology H = m

instance Real.pseudoMetricSpace : PseudoMetricSpace ℝ

Instantiate the reals as a pseudometric space.

Equations

• One or more equations did not get rendered due to their size.

theorem Real.dist_eq (x : ℝ) (y : ℝ) : dist x y = |x - y|

theorem Real.nndist_eq (x : ℝ) (y : ℝ) : nndist x y = Real.nnabs (x - y)

theorem Real.nndist_eq' (x : ℝ) (y : ℝ) : nndist x y = Real.nnabs (y - x)

theorem Real.dist_0_eq_abs (x : ℝ) : dist x 0 = |x|

theorem Real.sub_le_dist (x : ℝ) (y : ℝ) : x - y ≤ dist x y

theorem Real.ball_eq_Ioo (x : ℝ) (r : ℝ) : Metric.ball x r = Set.Ioo (x - r) (x + r)

theorem Real.closedBall_eq_Icc {x : ℝ} {r : ℝ} : Metric.closedBall x r = Set.Icc (x - r) (x + r)

theorem Real.Ioo_eq_ball (x : ℝ) (y : ℝ) : Set.Ioo x y = Metric.ball ((x + y) / 2) ((y - x) / 2)

theorem Real.Icc_eq_closedBall (x : ℝ) (y : ℝ) : Set.Icc x y = Metric.closedBall ((x + y) / 2) ((y - x) / 2)

theorem Metric.uniformity_eq_comap_nhds_zero {α : Type u} [PseudoMetricSpace α] : uniformity α = Filter.comap (fun (p : α × α) => dist p.1 p.2) (nhds 0)

theorem cauchySeq_iff_tendsto_dist_atTop_0 {α : Type u} {β : Type v} [PseudoMetricSpace α] [Nonempty β] [SemilatticeSup β] {u : β → α} : CauchySeq u ↔ Filter.Tendsto (fun (n : β × β) => dist (u n.1) (u n.2)) Filter.atTop (nhds 0)

theorem tendsto_uniformity_iff_dist_tendsto_zero {α : Type u} {ι : Type u_2} [PseudoMetricSpace α] {f : ι → α × α} {p : Filter ι} : Filter.Tendsto f p (uniformity α) ↔ Filter.Tendsto (fun (x : ι) => dist (f x).1 (f x).2) p (nhds 0)

theorem Filter.Tendsto.congr_dist {α : Type u} {ι : Type u_2} [PseudoMetricSpace α] {f₁ : ι → α} {f₂ : ι → α} {p : Filter ι} {a : α} (h₁ : Filter.Tendsto f₁ p (nhds a)) (h : Filter.Tendsto (fun (x : ι) => dist (f₁ x) (f₂ x)) p (nhds 0)) : Filter.Tendsto f₂ p (nhds a)

theorem tendsto_of_tendsto_of_dist {α : Type u} {ι : Type u_2} [PseudoMetricSpace α] {f₁ : ι → α} {f₂ : ι → α} {p : Filter ι} {a : α} (h₁ : Filter.Tendsto f₁ p (nhds a)) (h : Filter.Tendsto (fun (x : ι) => dist (f₁ x) (f₂ x)) p (nhds 0)) : Filter.Tendsto f₂ p (nhds a)

Alias of Filter.Tendsto.congr_dist.

theorem tendsto_iff_of_dist {α : Type u} {ι : Type u_2} [PseudoMetricSpace α] {f₁ : ι → α} {f₂ : ι → α} {p : Filter ι} {a : α} (h : Filter.Tendsto (fun (x : ι) => dist (f₁ x) (f₂ x)) p (nhds 0)) : Filter.Tendsto f₁ p (nhds a) ↔ Filter.Tendsto f₂ p (nhds a)

theorem dist_dist_dist_le_left {α : Type u} [PseudoMetricSpace α] (x : α) (y : α) (z : α) : dist (dist x z) (dist y z) ≤ dist x y

theorem dist_dist_dist_le_right {α : Type u} [PseudoMetricSpace α] (x : α) (y : α) (z : α) : dist (dist x y) (dist x z) ≤ dist y z

theorem dist_dist_dist_le {α : Type u} [PseudoMetricSpace α] (x : α) (y : α) (x' : α) (y' : α) : dist (dist x y) (dist x' y') ≤ dist x x' + dist y y'

theorem nhds_comap_dist {α : Type u} [PseudoMetricSpace α] (a : α) : Filter.comap (fun (x : α) => dist x a) (nhds 0) = nhds a

theorem tendsto_iff_dist_tendsto_zero {α : Type u} {β : Type v} [PseudoMetricSpace α] {f : β → α} {x : Filter β} {a : α} : Filter.Tendsto f x (nhds a) ↔ Filter.Tendsto (fun (b : β) => dist (f b) a) x (nhds 0)

theorem Metric.ball_subset_interior_closedBall {α : Type u} [PseudoMetricSpace α] {x : α} {ε : ℝ} : Metric.ball x ε ⊆ interior (Metric.closedBall x ε)

theorem Metric.mem_closure_iff {α : Type u} [PseudoMetricSpace α] {s : Set α} {a : α} : a ∈ closure s ↔ ∀ ε > 0, ∃ b ∈ s, dist a b < ε

ε-characterization of the closure in pseudometric spaces

theorem Metric.mem_closure_range_iff {α : Type u} {β : Type v} [PseudoMetricSpace α] {e : β → α} {a : α} : a ∈ closure (Set.range e) ↔ ∀ ε > 0, ∃ (k : β), dist a (e k) < ε

theorem Metric.mem_closure_range_iff_nat {α : Type u} {β : Type v} [PseudoMetricSpace α] {e : β → α} {a : α} : a ∈ closure (Set.range e) ↔ ∀ (n : ℕ), ∃ (k : β), dist a (e k) < 1 / (↑n + 1)

theorem Metric.mem_of_closed' {α : Type u} [PseudoMetricSpace α] {s : Set α} (hs : IsClosed s) {a : α} : a ∈ s ↔ ∀ ε > 0, ∃ b ∈ s, dist a b < ε

theorem Metric.dense_iff {α : Type u} [PseudoMetricSpace α] {s : Set α} : Dense s ↔ ∀ (x : α), ∀ r > 0, (Metric.ball x r ∩ s).Nonempty

theorem Metric.denseRange_iff {α : Type u} {β : Type v} [PseudoMetricSpace α] {f : β → α} : DenseRange f ↔ ∀ (x : α), ∀ r > 0, ∃ (y : β), dist x (f y) < r

theorem Inducing.isSeparable_preimage {α : Type u} {β : Type v} [PseudoMetricSpace α] {f : β → α} [TopologicalSpace β] (hf : Inducing f) {s : Set α} (hs : TopologicalSpace.IsSeparable s) : TopologicalSpace.IsSeparable (f ⁻¹' s)

The preimage of a separable set by an inducing map is separable.

theorem Embedding.isSeparable_preimage {α : Type u} {β : Type v} [PseudoMetricSpace α] {f : β → α} [TopologicalSpace β] (hf : Embedding f) {s : Set α} (hs : TopologicalSpace.IsSeparable s) : TopologicalSpace.IsSeparable (f ⁻¹' s)

theorem ContinuousOn.isSeparable_image {α : Type u} {β : Type v} [PseudoMetricSpace α] [TopologicalSpace β] {f : α → β} {s : Set α} (hf : ContinuousOn f s) (hs : TopologicalSpace.IsSeparable s) : TopologicalSpace.IsSeparable (f '' s)

If a map is continuous on a separable set s, then the image of s is also separable.

theorem IsCompact.isSeparable {α : Type u} [PseudoMetricSpace α] {s : Set α} (hs : IsCompact s) : TopologicalSpace.IsSeparable s

A compact set is separable.

theorem finite_cover_balls_of_compact {α : Type u} [PseudoMetricSpace α] {s : Set α} (hs : IsCompact s) {e : ℝ} (he : 0 < e) : ∃ t ⊆ s, t.Finite ∧ s ⊆ ⋃ x ∈ t, Metric.ball x e

Any compact set in a pseudometric space can be covered by finitely many balls of a given positive radius

theorem IsCompact.finite_cover_balls {α : Type u} [PseudoMetricSpace α] {s : Set α} (hs : IsCompact s) {e : ℝ} (he : 0 < e) : ∃ t ⊆ s, t.Finite ∧ s ⊆ ⋃ x ∈ t, Metric.ball x e

Alias of finite_cover_balls_of_compact.

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Any compact set in a pseudometric space can be covered by finitely many balls of a given positive radius

theorem Metric.secondCountable_of_almost_dense_set {α : Type u} [PseudoMetricSpace α] (H : ∀ ε > 0, ∃ (s : Set α), s.Countable ∧ ∀ (x : α), ∃ y ∈ s, dist x y ≤ ε) : SecondCountableTopology α

A pseudometric space is second countable if, for every ε > 0, there is a countable set which is ε-dense.

theorem lebesgue_number_lemma_of_metric {α : Type u} [PseudoMetricSpace α] {s : Set α} {ι : Sort u_3} {c : ι → Set α} (hs : IsCompact s) (hc₁ : ∀ (i : ι), IsOpen (c i)) (hc₂ : s ⊆ ⋃ (i : ι), c i) : ∃ δ > 0, ∀ x ∈ s, ∃ (i : ι), Metric.ball x δ ⊆ c i

theorem lebesgue_number_lemma_of_metric_sUnion {α : Type u} [PseudoMetricSpace α] {s : Set α} {c : Set (Set α)} (hs : IsCompact s) (hc₁ : ∀ t ∈ c, IsOpen t) (hc₂ : s ⊆ ⋃₀ c) : ∃ δ > 0, ∀ x ∈ s, ∃ t ∈ c, Metric.ball x δ ⊆ t