Extended metric spaces

This file is devoted to the definition and study of EMetricSpaces, i.e., metric spaces in which the distance is allowed to take the value ∞. This extended distance is called edist, and takes values in ℝ≥0∞.

Many definitions and theorems expected on emetric spaces are already introduced on uniform spaces and topological spaces. For example: open and closed sets, compactness, completeness, continuity and uniform continuity.

The class EMetricSpace therefore extends UniformSpace (and TopologicalSpace).

Since a lot of elementary properties don't require eq_of_edist_eq_zero we start setting up the theory of PseudoEMetricSpace, where we don't require edist x y = 0 → x = y and we specialize to EMetricSpace at the end.

theorem uniformity_dist_of_mem_uniformity {α : Type u} {β : Type v} [LinearOrder β] {U : Filter (α × α)} (z : β) (D : α → α → β) (H : ∀ (s : Set (α × α)), s ∈ U ↔ ∃ ε > z, ∀ {a b : α}, D a b < ε → (a, b) ∈ s) : U = ⨅ (ε : β), ⨅ (_ : ε > z), Filter.principal {p : α × α | D p.1 p.2 < ε}

Characterizing uniformities associated to a (generalized) distance function D in terms of the elements of the uniformity.

theorem EDist.ext {α : Type u_2} {x : EDist α} {y : EDist α} (edist : EDist.edist = EDist.edist) : x = y

theorem EDist.ext_iff {α : Type u_2} {x : EDist α} {y : EDist α} : x = y ↔ edist = edist

class EDist (α : Type u_2) : Type u_2

EDist α means that α is equipped with an extended distance.

• edist : α → α → ENNReal

def uniformSpaceOfEDist {α : Type u} (edist : α → α → ENNReal) (edist_self : ∀ (x : α), edist x x = 0) (edist_comm : ∀ (x y : α), edist x y = edist y x) (edist_triangle : ∀ (x y z : α), edist x z ≤ edist x y + edist y z) : UniformSpace α

Creating a uniform space from an extended distance.

Equations

• uniformSpaceOfEDist edist edist_self edist_comm edist_triangle = UniformSpace.ofFun edist edist_self edist_comm edist_triangle uniformSpaceOfEDist.proof_1

class PseudoEMetricSpace (α : Type u) extends EDist : Type u

Extended (pseudo) metric spaces, with an extended distance edist possibly taking the value ∞

Each pseudo_emetric 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 PseudoEMetricSpace structure, the uniformity fields are not necessary, they will be filled in by default. There is a default value for the uniformity, that can be substituted in cases of interest, for instance when instantiating a PseudoEMetricSpace structure on a product.

Continuity of edist is proved in Topology.Instances.ENNReal

• edist : α → α → ENNReal
• edist_self : ∀ (x : α), edist x x = 0
• edist_comm : ∀ (x y : α), edist x y = edist y x
• edist_triangle : ∀ (x y z : α), edist x z ≤ edist x y + edist y z
• toUniformSpace : UniformSpace α
• uniformity_edist : uniformity α = ⨅ (ε : ENNReal), ⨅ (_ : ε > 0), Filter.principal {p : α × α | edist p.1 p.2 < ε}

@[simp]

theorem PseudoEMetricSpace.edist_self {α : Type u} [self : PseudoEMetricSpace α] (x : α) : edist x x = 0

theorem PseudoEMetricSpace.edist_comm {α : Type u} [self : PseudoEMetricSpace α] (x : α) (y : α) : edist x y = edist y x

theorem PseudoEMetricSpace.edist_triangle {α : Type u} [self : PseudoEMetricSpace α] (x : α) (y : α) (z : α) : edist x z ≤ edist x y + edist y z

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

theorem PseudoEMetricSpace.ext_iff {α : Type u_2} {m : PseudoEMetricSpace α} {m' : PseudoEMetricSpace α} : m = m' ↔ PseudoEMetricSpace.toEDist = PseudoEMetricSpace.toEDist

theorem PseudoEMetricSpace.ext {α : Type u_2} {m : PseudoEMetricSpace α} {m' : PseudoEMetricSpace α} (h : PseudoEMetricSpace.toEDist = PseudoEMetricSpace.toEDist) : m = m'

Two pseudo emetric space structures with the same edistance function coincide.

theorem edist_triangle_left {α : Type u} [PseudoEMetricSpace α] (x : α) (y : α) (z : α) : edist x y ≤ edist z x + edist z y

Triangle inequality for the extended distance

theorem edist_triangle_right {α : Type u} [PseudoEMetricSpace α] (x : α) (y : α) (z : α) : edist x y ≤ edist x z + edist y z

theorem edist_congr_right {α : Type u} [PseudoEMetricSpace α] {x : α} {y : α} {z : α} (h : edist x y = 0) : edist x z = edist y z

theorem edist_congr_left {α : Type u} [PseudoEMetricSpace α] {x : α} {y : α} {z : α} (h : edist x y = 0) : edist z x = edist z y

theorem edist_congr {α : Type u} [PseudoEMetricSpace α] {w : α} {x : α} {y : α} {z : α} (hl : edist w x = 0) (hr : edist y z = 0) : edist w y = edist x z

theorem edist_triangle4 {α : Type u} [PseudoEMetricSpace α] (x : α) (y : α) (z : α) (t : α) : edist x t ≤ edist x y + edist y z + edist z t

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

Reformulation of the uniform structure in terms of the extended distance

theorem uniformSpace_edist {α : Type u} [PseudoEMetricSpace α] : PseudoEMetricSpace.toUniformSpace = uniformSpaceOfEDist edist ⋯ ⋯ ⋯

theorem uniformity_basis_edist {α : Type u} [PseudoEMetricSpace α] : (uniformity α).HasBasis (fun (ε : ENNReal) => 0 < ε) fun (ε : ENNReal) => {p : α × α | edist p.1 p.2 < ε}

theorem mem_uniformity_edist {α : Type u} [PseudoEMetricSpace α] {s : Set (α × α)} : s ∈ uniformity α ↔ ∃ ε > 0, ∀ {a b : α}, edist a b < ε → (a, b) ∈ s

Characterization of the elements of the uniformity in terms of the extended distance

theorem EMetric.mk_uniformity_basis {α : Type u} [PseudoEMetricSpace α] {β : Type u_2} {p : β → Prop} {f : β → ENNReal} (hf₀ : ∀ (x : β), p x → 0 < f x) (hf : ∀ (ε : ENNReal), 0 < ε → ∃ (x : β), p x ∧ f x ≤ ε) : (uniformity α).HasBasis p fun (x : β) => {p : α × α | edist p.1 p.2 < f x}

Given f : β → ℝ≥0∞, 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_edist, uniformity_basis_edist', uniformity_basis_edist_nnreal, and uniformity_basis_edist_inv_nat.

theorem EMetric.mk_uniformity_basis_le {α : Type u} [PseudoEMetricSpace α] {β : Type u_2} {p : β → Prop} {f : β → ENNReal} (hf₀ : ∀ (x : β), p x → 0 < f x) (hf : ∀ (ε : ENNReal), 0 < ε → ∃ (x : β), p x ∧ f x ≤ ε) : (uniformity α).HasBasis p fun (x : β) => {p : α × α | edist p.1 p.2 ≤ f x}

Given f : β → ℝ≥0∞, if f sends {i | p i} to a set of positive numbers accumulating to zero, then closed f i-neighborhoods of the diagonal form a basis of 𝓤 α.

For specific bases see uniformity_basis_edist_le and uniformity_basis_edist_le'.

theorem uniformity_basis_edist_le {α : Type u} [PseudoEMetricSpace α] : (uniformity α).HasBasis (fun (ε : ENNReal) => 0 < ε) fun (ε : ENNReal) => {p : α × α | edist p.1 p.2 ≤ ε}

theorem uniformity_basis_edist' {α : Type u} [PseudoEMetricSpace α] (ε' : ENNReal) (hε' : 0 < ε') : (uniformity α).HasBasis (fun (ε : ENNReal) => ε ∈ Set.Ioo 0 ε') fun (ε : ENNReal) => {p : α × α | edist p.1 p.2 < ε}

theorem uniformity_basis_edist_le' {α : Type u} [PseudoEMetricSpace α] (ε' : ENNReal) (hε' : 0 < ε') : (uniformity α).HasBasis (fun (ε : ENNReal) => ε ∈ Set.Ioo 0 ε') fun (ε : ENNReal) => {p : α × α | edist p.1 p.2 ≤ ε}

theorem uniformity_basis_edist_nnreal {α : Type u} [PseudoEMetricSpace α] : (uniformity α).HasBasis (fun (ε : NNReal) => 0 < ε) fun (ε : NNReal) => {p : α × α | edist p.1 p.2 < ↑ε}

theorem uniformity_basis_edist_nnreal_le {α : Type u} [PseudoEMetricSpace α] : (uniformity α).HasBasis (fun (ε : NNReal) => 0 < ε) fun (ε : NNReal) => {p : α × α | edist p.1 p.2 ≤ ↑ε}

theorem uniformity_basis_edist_inv_nat {α : Type u} [PseudoEMetricSpace α] : (uniformity α).HasBasis (fun (x : ℕ) => True) fun (n : ℕ) => {p : α × α | edist p.1 p.2 < (↑n)⁻¹}

theorem uniformity_basis_edist_inv_two_pow {α : Type u} [PseudoEMetricSpace α] : (uniformity α).HasBasis (fun (x : ℕ) => True) fun (n : ℕ) => {p : α × α | edist p.1 p.2 < 2⁻¹ ^ n}

theorem edist_mem_uniformity {α : Type u} [PseudoEMetricSpace α] {ε : ENNReal} (ε0 : 0 < ε) : {p : α × α | edist p.1 p.2 < ε} ∈ uniformity α

Fixed size neighborhoods of the diagonal belong to the uniform structure

@[instance 900]

instance EMetric.instIsCountablyGeneratedUniformity {α : Type u} [PseudoEMetricSpace α] : (uniformity α).IsCountablyGenerated

Equations

• ⋯ = ⋯

theorem EMetric.uniformContinuousOn_iff {α : Type u} {β : Type v} [PseudoEMetricSpace α] [PseudoEMetricSpace β] {f : α → β} {s : Set α} : UniformContinuousOn f s ↔ ∀ ε > 0, ∃ δ > 0, ∀ {a : α}, a ∈ s → ∀ {b : α}, b ∈ s → edist a b < δ → edist (f a) (f b) < ε

ε-δ characterization of uniform continuity on a set for pseudoemetric spaces

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

ε-δ characterization of uniform continuity on pseudoemetric spaces

@[reducible, inline]

abbrev PseudoEMetricSpace.replaceUniformity {α : Type u_2} [U : UniformSpace α] (m : PseudoEMetricSpace α) (H : uniformity α = uniformity α) : PseudoEMetricSpace α

Auxiliary function to replace the uniformity on a pseudoemetric space with a uniformity which is equal to the original one, but maybe not defeq. This is useful if one wants to construct a pseudoemetric space with a specified uniformity. See Note [forgetful inheritance] explaining why having definitionally the right uniformity is often important. See note [reducible non-instances].

Equations

• m.replaceUniformity H = PseudoEMetricSpace.mk ⋯ ⋯ ⋯ U ⋯

@[reducible, inline]

abbrev PseudoEMetricSpace.induced {α : Type u_2} {β : Type u_3} (f : α → β) (m : PseudoEMetricSpace β) : PseudoEMetricSpace α

The extended pseudometric induced by a function taking values in a pseudoemetric space. See note [reducible non-instances].

Equations

• PseudoEMetricSpace.induced f m = PseudoEMetricSpace.mk ⋯ ⋯ ⋯ (UniformSpace.comap f PseudoEMetricSpace.toUniformSpace) ⋯

instance instPseudoEMetricSpaceSubtype {α : Type u_2} {p : α → Prop} [PseudoEMetricSpace α] : PseudoEMetricSpace (Subtype p)

Pseudoemetric space instance on subsets of pseudoemetric spaces

Equations

• instPseudoEMetricSpaceSubtype = PseudoEMetricSpace.induced Subtype.val inst

theorem Subtype.edist_eq {α : Type u} [PseudoEMetricSpace α] {p : α → Prop} (x : Subtype p) (y : Subtype p) : edist x y = edist ↑x ↑y

The extended pseudodistance on a subset of a pseudoemetric space is the restriction of the original pseudodistance, by definition

instance AddOpposite.instPseudoEMetricSpace {α : Type u_2} [PseudoEMetricSpace α] : PseudoEMetricSpace αᵃᵒᵖ

Pseudoemetric space instance on the additive opposite of a pseudoemetric space.

Equations

• AddOpposite.instPseudoEMetricSpace = PseudoEMetricSpace.induced AddOpposite.unop inst

instance MulOpposite.instPseudoEMetricSpace {α : Type u_2} [PseudoEMetricSpace α] : PseudoEMetricSpace αᵐᵒᵖ

Pseudoemetric space instance on the multiplicative opposite of a pseudoemetric space.

Equations

• MulOpposite.instPseudoEMetricSpace = PseudoEMetricSpace.induced MulOpposite.unop inst

theorem AddOpposite.edist_unop {α : Type u} [PseudoEMetricSpace α] (x : αᵃᵒᵖ) (y : αᵃᵒᵖ) : edist (AddOpposite.unop x) (AddOpposite.unop y) = edist x y

theorem MulOpposite.edist_unop {α : Type u} [PseudoEMetricSpace α] (x : αᵐᵒᵖ) (y : αᵐᵒᵖ) : edist (MulOpposite.unop x) (MulOpposite.unop y) = edist x y

theorem AddOpposite.edist_op {α : Type u} [PseudoEMetricSpace α] (x : α) (y : α) : edist (AddOpposite.op x) (AddOpposite.op y) = edist x y

theorem MulOpposite.edist_op {α : Type u} [PseudoEMetricSpace α] (x : α) (y : α) : edist (MulOpposite.op x) (MulOpposite.op y) = edist x y

instance instPseudoEMetricSpaceULift {α : Type u} [PseudoEMetricSpace α] : PseudoEMetricSpace (ULift.{u_2, u} α)

Equations

• instPseudoEMetricSpaceULift = PseudoEMetricSpace.induced ULift.down inst

theorem ULift.edist_eq {α : Type u} [PseudoEMetricSpace α] (x : ULift.{u_2, u} α) (y : ULift.{u_2, u} α) : edist x y = edist x.down y.down

@[simp]

theorem ULift.edist_up_up {α : Type u} [PseudoEMetricSpace α] (x : α) (y : α) : edist { down := x } { down := y } = edist x y

instance Prod.pseudoEMetricSpaceMax {α : Type u} {β : Type v} [PseudoEMetricSpace α] [PseudoEMetricSpace β] : PseudoEMetricSpace (α × β)

The product of two pseudoemetric spaces, with the max distance, is an extended pseudometric spaces. We make sure that the uniform structure thus constructed is the one corresponding to the product of uniform spaces, to avoid diamond problems.

Equations

• Prod.pseudoEMetricSpaceMax = PseudoEMetricSpace.mk ⋯ ⋯ ⋯ inferInstance ⋯

theorem Prod.edist_eq {α : Type u} {β : Type v} [PseudoEMetricSpace α] [PseudoEMetricSpace β] (x : α × β) (y : α × β) : edist x y = max (edist x.1 y.1) (edist x.2 y.2)

def EMetric.ball {α : Type u} [PseudoEMetricSpace α] (x : α) (ε : ENNReal) : Set α

EMetric.ball x ε is the set of all points y with edist y x < ε

Equations

• EMetric.ball x ε = {y : α | edist y x < ε}

@[simp]

theorem EMetric.mem_ball {α : Type u} [PseudoEMetricSpace α] {x : α} {y : α} {ε : ENNReal} : y ∈ EMetric.ball x ε ↔ edist y x < ε

theorem EMetric.mem_ball' {α : Type u} [PseudoEMetricSpace α] {x : α} {y : α} {ε : ENNReal} : y ∈ EMetric.ball x ε ↔ edist x y < ε

def EMetric.closedBall {α : Type u} [PseudoEMetricSpace α] (x : α) (ε : ENNReal) : Set α

EMetric.closedBall x ε is the set of all points y with edist y x ≤ ε

Equations

• EMetric.closedBall x ε = {y : α | edist y x ≤ ε}

@[simp]

theorem EMetric.mem_closedBall {α : Type u} [PseudoEMetricSpace α] {x : α} {y : α} {ε : ENNReal} : y ∈ EMetric.closedBall x ε ↔ edist y x ≤ ε

theorem EMetric.mem_closedBall' {α : Type u} [PseudoEMetricSpace α] {x : α} {y : α} {ε : ENNReal} : y ∈ EMetric.closedBall x ε ↔ edist x y ≤ ε

@[simp]

theorem EMetric.closedBall_top {α : Type u} [PseudoEMetricSpace α] (x : α) : EMetric.closedBall x ⊤ = Set.univ

theorem EMetric.ball_subset_closedBall {α : Type u} [PseudoEMetricSpace α] {x : α} {ε : ENNReal} : EMetric.ball x ε ⊆ EMetric.closedBall x ε

theorem EMetric.pos_of_mem_ball {α : Type u} [PseudoEMetricSpace α] {x : α} {y : α} {ε : ENNReal} (hy : y ∈ EMetric.ball x ε) : 0 < ε

theorem EMetric.mem_ball_self {α : Type u} [PseudoEMetricSpace α] {x : α} {ε : ENNReal} (h : 0 < ε) : x ∈ EMetric.ball x ε

theorem EMetric.mem_closedBall_self {α : Type u} [PseudoEMetricSpace α] {x : α} {ε : ENNReal} : x ∈ EMetric.closedBall x ε

theorem EMetric.mem_ball_comm {α : Type u} [PseudoEMetricSpace α] {x : α} {y : α} {ε : ENNReal} : x ∈ EMetric.ball y ε ↔ y ∈ EMetric.ball x ε

theorem EMetric.mem_closedBall_comm {α : Type u} [PseudoEMetricSpace α] {x : α} {y : α} {ε : ENNReal} : x ∈ EMetric.closedBall y ε ↔ y ∈ EMetric.closedBall x ε

theorem EMetric.ball_subset_ball {α : Type u} [PseudoEMetricSpace α] {x : α} {ε₁ : ENNReal} {ε₂ : ENNReal} (h : ε₁ ≤ ε₂) : EMetric.ball x ε₁ ⊆ EMetric.ball x ε₂

theorem EMetric.closedBall_subset_closedBall {α : Type u} [PseudoEMetricSpace α] {x : α} {ε₁ : ENNReal} {ε₂ : ENNReal} (h : ε₁ ≤ ε₂) : EMetric.closedBall x ε₁ ⊆ EMetric.closedBall x ε₂

theorem EMetric.ball_disjoint {α : Type u} [PseudoEMetricSpace α] {x : α} {y : α} {ε₁ : ENNReal} {ε₂ : ENNReal} (h : ε₁ + ε₂ ≤ edist x y) : Disjoint (EMetric.ball x ε₁) (EMetric.ball y ε₂)

theorem EMetric.ball_subset {α : Type u} [PseudoEMetricSpace α] {x : α} {y : α} {ε₁ : ENNReal} {ε₂ : ENNReal} (h : edist x y + ε₁ ≤ ε₂) (h' : edist x y ≠ ⊤) : EMetric.ball x ε₁ ⊆ EMetric.ball y ε₂

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

theorem EMetric.ball_eq_empty_iff {α : Type u} [PseudoEMetricSpace α] {x : α} {ε : ENNReal} : EMetric.ball x ε = ∅ ↔ ε = 0

theorem EMetric.ordConnected_setOf_closedBall_subset {α : Type u} [PseudoEMetricSpace α] (x : α) (s : Set α) : {r : ENNReal | EMetric.closedBall x r ⊆ s}.OrdConnected

theorem EMetric.ordConnected_setOf_ball_subset {α : Type u} [PseudoEMetricSpace α] (x : α) (s : Set α) : {r : ENNReal | EMetric.ball x r ⊆ s}.OrdConnected

def EMetric.edistLtTopSetoid {α : Type u} [PseudoEMetricSpace α] : Setoid α

Relation “two points are at a finite edistance” is an equivalence relation.

Equations

• EMetric.edistLtTopSetoid = { r := fun (x y : α) => edist x y < ⊤, iseqv := ⋯ }

@[simp]

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

theorem EMetric.nhds_basis_eball {α : Type u} [PseudoEMetricSpace α] {x : α} : (nhds x).HasBasis (fun (ε : ENNReal) => 0 < ε) (EMetric.ball x)

theorem EMetric.nhdsWithin_basis_eball {α : Type u} [PseudoEMetricSpace α] {x : α} {s : Set α} : (nhdsWithin x s).HasBasis (fun (ε : ENNReal) => 0 < ε) fun (ε : ENNReal) => EMetric.ball x ε ∩ s

theorem EMetric.nhds_basis_closed_eball {α : Type u} [PseudoEMetricSpace α] {x : α} : (nhds x).HasBasis (fun (ε : ENNReal) => 0 < ε) (EMetric.closedBall x)

theorem EMetric.nhdsWithin_basis_closed_eball {α : Type u} [PseudoEMetricSpace α] {x : α} {s : Set α} : (nhdsWithin x s).HasBasis (fun (ε : ENNReal) => 0 < ε) fun (ε : ENNReal) => EMetric.closedBall x ε ∩ s

theorem EMetric.nhds_eq {α : Type u} [PseudoEMetricSpace α] {x : α} : nhds x = ⨅ (ε : ENNReal), ⨅ (_ : ε > 0), Filter.principal (EMetric.ball x ε)

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

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

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

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

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

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

theorem EMetric.isOpen_ball {α : Type u} [PseudoEMetricSpace α] {x : α} {ε : ENNReal} : IsOpen (EMetric.ball x ε)

theorem EMetric.isClosed_ball_top {α : Type u} [PseudoEMetricSpace α] {x : α} : IsClosed (EMetric.ball x ⊤)

theorem EMetric.ball_mem_nhds {α : Type u} [PseudoEMetricSpace α] (x : α) {ε : ENNReal} (ε0 : 0 < ε) : EMetric.ball x ε ∈ nhds x

theorem EMetric.closedBall_mem_nhds {α : Type u} [PseudoEMetricSpace α] (x : α) {ε : ENNReal} (ε0 : 0 < ε) : EMetric.closedBall x ε ∈ nhds x

theorem EMetric.ball_prod_same {α : Type u} {β : Type v} [PseudoEMetricSpace α] [PseudoEMetricSpace β] (x : α) (y : β) (r : ENNReal) : EMetric.ball x r ×ˢ EMetric.ball y r = EMetric.ball (x, y) r

theorem EMetric.closedBall_prod_same {α : Type u} {β : Type v} [PseudoEMetricSpace α] [PseudoEMetricSpace β] (x : α) (y : β) (r : ENNReal) : EMetric.closedBall x r ×ˢ EMetric.closedBall y r = EMetric.closedBall (x, y) r

theorem EMetric.mem_closure_iff {α : Type u} [PseudoEMetricSpace α] {x : α} {s : Set α} : x ∈ closure s ↔ ∀ ε > 0, ∃ y ∈ s, edist x y < ε

ε-characterization of the closure in pseudoemetric spaces

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

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

theorem EMetric.subset_countable_closure_of_almost_dense_set {α : Type u} [PseudoEMetricSpace α] (s : Set α) (hs : ∀ ε > 0, ∃ (t : Set α), t.Countable ∧ s ⊆ ⋃ x ∈ t, EMetric.closedBall x ε) : ∃ t ⊆ s, t.Countable ∧ s ⊆ closure t

For a set s in a pseudo emetric space, if for every ε > 0 there exists a countable set that is ε-dense in s, then there exists a countable subset t ⊆ s that is dense in s.

theorem TopologicalSpace.IsSeparable.exists_countable_dense_subset {α : Type u} [PseudoEMetricSpace α] {s : Set α} (hs : TopologicalSpace.IsSeparable s) : ∃ t ⊆ s, t.Countable ∧ s ⊆ closure t

If a set s is separable in a (pseudo extended) metric space, then it admits a countable dense subset. This is not obvious, as the countable set whose closure covers s given by the definition of separability does not need in general to be contained in s.

theorem TopologicalSpace.IsSeparable.separableSpace {α : Type u} [PseudoEMetricSpace α] {s : Set α} (hs : TopologicalSpace.IsSeparable s) : TopologicalSpace.SeparableSpace ↑s

If a set s is separable, then the corresponding subtype is separable in a (pseudo extended) metric space. This is not obvious, as the countable set whose closure covers s does not need in general to be contained in s.

class EMetricSpace (α : Type u) extends PseudoEMetricSpace : Type u

We now define EMetricSpace, extending PseudoEMetricSpace.

• edist : α → α → ENNReal
• edist_self : ∀ (x : α), edist x x = 0
• edist_comm : ∀ (x y : α), edist x y = edist y x
• edist_triangle : ∀ (x y z : α), edist x z ≤ edist x y + edist y z
• toUniformSpace : UniformSpace α
• uniformity_edist : uniformity α = ⨅ (ε : ENNReal), ⨅ (_ : ε > 0), Filter.principal {p : α × α | edist p.1 p.2 < ε}
• eq_of_edist_eq_zero : ∀ {x y : α}, edist x y = 0 → x = y

theorem EMetricSpace.eq_of_edist_eq_zero {α : Type u} [self : EMetricSpace α] {x : α} {y : α} : edist x y = 0 → x = y

theorem EMetricSpace.ext_iff {α : Type u_2} {m : EMetricSpace α} {m' : EMetricSpace α} : m = m' ↔ PseudoEMetricSpace.toEDist = PseudoEMetricSpace.toEDist

theorem EMetricSpace.ext {α : Type u_2} {m : EMetricSpace α} {m' : EMetricSpace α} (h : PseudoEMetricSpace.toEDist = PseudoEMetricSpace.toEDist) : m = m'

@[simp]

theorem edist_eq_zero {γ : Type w} [EMetricSpace γ] {x : γ} {y : γ} : edist x y = 0 ↔ x = y

Characterize the equality of points by the vanishing of their extended distance

@[simp]

theorem zero_eq_edist {γ : Type w} [EMetricSpace γ] {x : γ} {y : γ} : 0 = edist x y ↔ x = y

theorem edist_le_zero {γ : Type w} [EMetricSpace γ] {x : γ} {y : γ} : edist x y ≤ 0 ↔ x = y

@[simp]

theorem edist_pos {γ : Type w} [EMetricSpace γ] {x : γ} {y : γ} : 0 < edist x y ↔ x ≠ y

theorem eq_of_forall_edist_le {γ : Type w} [EMetricSpace γ] {x : γ} {y : γ} (h : ∀ ε > 0, edist x y ≤ ε) : x = y

Two points coincide if their distance is < ε for all positive ε

@[reducible, inline]

abbrev EMetricSpace.replaceUniformity {γ : Type u_2} [U : UniformSpace γ] (m : EMetricSpace γ) (H : uniformity γ = uniformity γ) : EMetricSpace γ

Auxiliary function to replace the uniformity on an emetric space with a uniformity which is equal to the original one, but maybe not defeq. This is useful if one wants to construct an emetric space with a specified uniformity. See Note [forgetful inheritance] explaining why having definitionally the right uniformity is often important. See note [reducible non-instances].

Equations

• m.replaceUniformity H = EMetricSpace.mk ⋯

@[reducible, inline]

abbrev EMetricSpace.induced {γ : Type u_2} {β : Type u_3} (f : γ → β) (hf : Function.Injective f) (m : EMetricSpace β) : EMetricSpace γ

The extended metric induced by an injective function taking values in an emetric space. See Note [reducible non-instances].

Equations

• EMetricSpace.induced f hf m = let __src := PseudoEMetricSpace.induced f EMetricSpace.toPseudoEMetricSpace; EMetricSpace.mk ⋯

instance instEMetricSpaceSubtype {α : Type u_2} {p : α → Prop} [EMetricSpace α] : EMetricSpace (Subtype p)

EMetric space instance on subsets of emetric spaces

Equations

• instEMetricSpaceSubtype = EMetricSpace.induced Subtype.val ⋯ inst

instance instEMetricSpaceAddOpposite {α : Type u_2} [EMetricSpace α] : EMetricSpace αᵃᵒᵖ

EMetric space instance on the additive opposite of an emetric space.

Equations

• instEMetricSpaceAddOpposite = EMetricSpace.induced AddOpposite.unop ⋯ inst

instance instEMetricSpaceMulOpposite {α : Type u_2} [EMetricSpace α] : EMetricSpace αᵐᵒᵖ

EMetric space instance on the multiplicative opposite of an emetric space.

Equations

• instEMetricSpaceMulOpposite = EMetricSpace.induced MulOpposite.unop ⋯ inst

instance instEMetricSpaceULift {α : Type u_2} [EMetricSpace α] : EMetricSpace (ULift.{u_3, u_2} α)

Equations

• instEMetricSpaceULift = EMetricSpace.induced ULift.down ⋯ inst

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

Reformulation of the uniform structure in terms of the extended distance

Additive, Multiplicative

The distance on those type synonyms is inherited without change.

instance instEDistAdditive {X : Type u_1} [EDist X] : EDist (Additive X)

Equations

• instEDistAdditive = inst

instance instEDistMultiplicative {X : Type u_1} [EDist X] : EDist (Multiplicative X)

Equations

• instEDistMultiplicative = inst

@[simp]

theorem edist_ofMul {X : Type u_1} [EDist X] (a : X) (b : X) : edist (Additive.ofMul a) (Additive.ofMul b) = edist a b

@[simp]

theorem edist_ofAdd {X : Type u_1} [EDist X] (a : X) (b : X) : edist (Multiplicative.ofAdd a) (Multiplicative.ofAdd b) = edist a b

@[simp]

theorem edist_toMul {X : Type u_1} [EDist X] (a : Additive X) (b : Additive X) : edist (Additive.toMul a) (Additive.toMul b) = edist a b

@[simp]

theorem edist_toAdd {X : Type u_1} [EDist X] (a : Multiplicative X) (b : Multiplicative X) : edist (Multiplicative.toAdd a) (Multiplicative.toAdd b) = edist a b

instance instPseudoEMetricSpaceAdditive {X : Type u_1} [PseudoEMetricSpace X] : PseudoEMetricSpace (Additive X)

Equations

• instPseudoEMetricSpaceAdditive = inst

instance instPseudoEMetricSpaceMultiplicative {X : Type u_1} [PseudoEMetricSpace X] : PseudoEMetricSpace (Multiplicative X)

Equations

• instPseudoEMetricSpaceMultiplicative = inst

instance instEMetricSpaceAdditive {X : Type u_1} [EMetricSpace X] : EMetricSpace (Additive X)

Equations

• instEMetricSpaceAdditive = inst

instance instEMetricSpaceMultiplicative {X : Type u_1} [EMetricSpace X] : EMetricSpace (Multiplicative X)

Equations

• instEMetricSpaceMultiplicative = inst

Order dual

The distance on this type synonym is inherited without change.

instance instEDistOrderDual {X : Type u_1} [EDist X] : EDist Xᵒᵈ

Equations

• instEDistOrderDual = inst

@[simp]

theorem edist_toDual {X : Type u_1} [EDist X] (a : X) (b : X) : edist (OrderDual.toDual a) (OrderDual.toDual b) = edist a b

@[simp]

theorem edist_ofDual {X : Type u_1} [EDist X] (a : Xᵒᵈ) (b : Xᵒᵈ) : edist (OrderDual.ofDual a) (OrderDual.ofDual b) = edist a b

instance instPseudoEMetricSpaceOrderDual {X : Type u_1} [PseudoEMetricSpace X] : PseudoEMetricSpace Xᵒᵈ

Equations

• instPseudoEMetricSpaceOrderDual = inst

instance instEMetricSpaceOrderDual {X : Type u_1} [EMetricSpace X] : EMetricSpace Xᵒᵈ

Equations

• instEMetricSpaceOrderDual = inst