fact stringlengths 6 3.84k | type stringclasses 11 values | library stringclasses 32 values | imports listlengths 1 14 | filename stringlengths 20 95 | symbolic_name stringlengths 1 90 | docstring stringlengths 7 20k ⌀ |
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@[to_additive /-- Pointwise, the indicators of δ-thickenings of a set eventually coincide
with the indicator of the set as δ>0 tends to zero. -/]
mulIndicator_thickening_eventually_eq_mulIndicator_closure (f : α → β) (E : Set α) (x : α) :
∀ᶠ δ in 𝓝[>] (0 : ℝ),
(Metric.thickening δ E).mulIndicator f x = (closure E).mulIndicator f x := by
by_cases x_mem_closure : x ∈ closure E
· filter_upwards [self_mem_nhdsWithin] with δ δ_pos
simp only [closure_subset_thickening δ_pos E x_mem_closure, mulIndicator_of_mem, x_mem_closure]
· have obs := eventually_notMem_thickening_of_infEdist_pos x_mem_closure
filter_upwards [mem_nhdsWithin_of_mem_nhds obs, self_mem_nhdsWithin]
with δ x_notin_thE _
simp only [x_notin_thE, not_false_eq_true, mulIndicator_of_notMem, x_mem_closure] | lemma | Topology | [
"Mathlib.Data.ENNReal.Lemmas",
"Mathlib.Topology.MetricSpace.Thickening",
"Mathlib.Topology.ContinuousMap.Bounded.Basic"
] | Mathlib/Topology/MetricSpace/ThickenedIndicator.lean | mulIndicator_thickening_eventually_eq_mulIndicator_closure | Pointwise, the multiplicative indicators of δ-thickenings of a set eventually coincide
with the multiplicative indicator of the set as δ>0 tends to zero. |
@[to_additive /-- Pointwise, the indicators of closed δ-thickenings of a set eventually coincide
with the indicator of the set as δ tends to zero. -/]
mulIndicator_cthickening_eventually_eq_mulIndicator_closure (f : α → β) (E : Set α) (x : α) :
∀ᶠ δ in 𝓝 (0 : ℝ),
(Metric.cthickening δ E).mulIndicator f x = (closure E).mulIndicator f x := by
by_cases x_mem_closure : x ∈ closure E
· filter_upwards [univ_mem] with δ _
have obs : x ∈ cthickening δ E := closure_subset_cthickening δ E x_mem_closure
rw [mulIndicator_of_mem obs f, mulIndicator_of_mem x_mem_closure f]
· filter_upwards [eventually_notMem_cthickening_of_infEdist_pos x_mem_closure] with δ hδ
simp only [hδ, not_false_eq_true, mulIndicator_of_notMem, x_mem_closure]
variable [TopologicalSpace β] | lemma | Topology | [
"Mathlib.Data.ENNReal.Lemmas",
"Mathlib.Topology.MetricSpace.Thickening",
"Mathlib.Topology.ContinuousMap.Bounded.Basic"
] | Mathlib/Topology/MetricSpace/ThickenedIndicator.lean | mulIndicator_cthickening_eventually_eq_mulIndicator_closure | Pointwise, the multiplicative indicators of closed δ-thickenings of a set eventually coincide
with the multiplicative indicator of the set as δ tends to zero. |
@[to_additive /-- The indicators of δ-thickenings of a set tend pointwise to the indicator of the
set, as δ>0 tends to zero. -/]
tendsto_mulIndicator_thickening_mulIndicator_closure (f : α → β) (E : Set α) :
Tendsto (fun δ ↦ (Metric.thickening δ E).mulIndicator f) (𝓝[>] 0)
(𝓝 ((closure E).mulIndicator f)) := by
rw [tendsto_pi_nhds]
intro x
rw [tendsto_congr' (mulIndicator_thickening_eventually_eq_mulIndicator_closure f E x)]
apply tendsto_const_nhds | lemma | Topology | [
"Mathlib.Data.ENNReal.Lemmas",
"Mathlib.Topology.MetricSpace.Thickening",
"Mathlib.Topology.ContinuousMap.Bounded.Basic"
] | Mathlib/Topology/MetricSpace/ThickenedIndicator.lean | tendsto_mulIndicator_thickening_mulIndicator_closure | The multiplicative indicators of δ-thickenings of a set tend pointwise to the multiplicative
indicator of the set, as δ>0 tends to zero. |
@[to_additive /-- The indicators of closed δ-thickenings of a set tend pointwise to the indicator
of the set, as δ tends to zero. -/]
tendsto_mulIndicator_cthickening_mulIndicator_closure (f : α → β) (E : Set α) :
Tendsto (fun δ ↦ (Metric.cthickening δ E).mulIndicator f) (𝓝 0)
(𝓝 ((closure E).mulIndicator f)) := by
rw [tendsto_pi_nhds]
intro x
rw [tendsto_congr' (mulIndicator_cthickening_eventually_eq_mulIndicator_closure f E x)]
apply tendsto_const_nhds | lemma | Topology | [
"Mathlib.Data.ENNReal.Lemmas",
"Mathlib.Topology.MetricSpace.Thickening",
"Mathlib.Topology.ContinuousMap.Bounded.Basic"
] | Mathlib/Topology/MetricSpace/ThickenedIndicator.lean | tendsto_mulIndicator_cthickening_mulIndicator_closure | The multiplicative indicators of closed δ-thickenings of a set tend pointwise to the
multiplicative indicator of the set, as δ tends to zero. |
thickening (δ : ℝ) (E : Set α) : Set α :=
{ x : α | infEdist x E < ENNReal.ofReal δ } | def | Topology | [
"Mathlib.Topology.MetricSpace.HausdorffDistance"
] | Mathlib/Topology/MetricSpace/Thickening.lean | thickening | The (open) `δ`-thickening `Metric.thickening δ E` of a subset `E` in a pseudo emetric space
consists of those points that are at distance less than `δ` from some point of `E`. |
mem_thickening_iff_infEdist_lt : x ∈ thickening δ s ↔ infEdist x s < ENNReal.ofReal δ :=
Iff.rfl | theorem | Topology | [
"Mathlib.Topology.MetricSpace.HausdorffDistance"
] | Mathlib/Topology/MetricSpace/Thickening.lean | mem_thickening_iff_infEdist_lt | null |
eventually_notMem_thickening_of_infEdist_pos {E : Set α} {x : α} (h : x ∉ closure E) :
∀ᶠ δ in 𝓝 (0 : ℝ), x ∉ Metric.thickening δ E := by
obtain ⟨ε, ⟨ε_pos, ε_lt⟩⟩ := exists_real_pos_lt_infEdist_of_notMem_closure h
filter_upwards [eventually_lt_nhds ε_pos] with δ hδ
simp only [thickening, mem_setOf_eq, not_lt]
exact (ENNReal.ofReal_le_ofReal hδ.le).trans ε_lt.le
@[deprecated (since := "2025-05-23")]
alias eventually_not_mem_thickening_of_infEdist_pos := eventually_notMem_thickening_of_infEdist_pos | lemma | Topology | [
"Mathlib.Topology.MetricSpace.HausdorffDistance"
] | Mathlib/Topology/MetricSpace/Thickening.lean | eventually_notMem_thickening_of_infEdist_pos | An exterior point of a subset `E` (i.e., a point outside the closure of `E`) is not in the
(open) `δ`-thickening of `E` for small enough positive `δ`. |
thickening_eq_preimage_infEdist (δ : ℝ) (E : Set α) :
thickening δ E = (infEdist · E) ⁻¹' Iio (ENNReal.ofReal δ) :=
rfl | theorem | Topology | [
"Mathlib.Topology.MetricSpace.HausdorffDistance"
] | Mathlib/Topology/MetricSpace/Thickening.lean | thickening_eq_preimage_infEdist | The (open) thickening equals the preimage of an open interval under `EMetric.infEdist`. |
isOpen_thickening {δ : ℝ} {E : Set α} : IsOpen (thickening δ E) :=
Continuous.isOpen_preimage continuous_infEdist _ isOpen_Iio | theorem | Topology | [
"Mathlib.Topology.MetricSpace.HausdorffDistance"
] | Mathlib/Topology/MetricSpace/Thickening.lean | isOpen_thickening | The (open) thickening is an open set. |
@[simp]
thickening_empty (δ : ℝ) : thickening δ (∅ : Set α) = ∅ := by
simp only [thickening, setOf_false, infEdist_empty, not_top_lt] | theorem | Topology | [
"Mathlib.Topology.MetricSpace.HausdorffDistance"
] | Mathlib/Topology/MetricSpace/Thickening.lean | thickening_empty | The (open) thickening of the empty set is empty. |
thickening_of_nonpos (hδ : δ ≤ 0) (s : Set α) : thickening δ s = ∅ :=
eq_empty_of_forall_notMem fun _ => ((ENNReal.ofReal_of_nonpos hδ).trans_le bot_le).not_gt | theorem | Topology | [
"Mathlib.Topology.MetricSpace.HausdorffDistance"
] | Mathlib/Topology/MetricSpace/Thickening.lean | thickening_of_nonpos | null |
@[gcongr]
thickening_mono {δ₁ δ₂ : ℝ} (hle : δ₁ ≤ δ₂) (E : Set α) :
thickening δ₁ E ⊆ thickening δ₂ E :=
preimage_mono (Iio_subset_Iio (ENNReal.ofReal_le_ofReal hle)) | theorem | Topology | [
"Mathlib.Topology.MetricSpace.HausdorffDistance"
] | Mathlib/Topology/MetricSpace/Thickening.lean | thickening_mono | The (open) thickening `Metric.thickening δ E` of a fixed subset `E` is an increasing function of
the thickening radius `δ`. |
thickening_subset_of_subset (δ : ℝ) {E₁ E₂ : Set α} (h : E₁ ⊆ E₂) :
thickening δ E₁ ⊆ thickening δ E₂ := fun _ hx => lt_of_le_of_lt (infEdist_anti h) hx | theorem | Topology | [
"Mathlib.Topology.MetricSpace.HausdorffDistance"
] | Mathlib/Topology/MetricSpace/Thickening.lean | thickening_subset_of_subset | The (open) thickening `Metric.thickening δ E` with a fixed thickening radius `δ` is
an increasing function of the subset `E`. |
mem_thickening_iff_exists_edist_lt {δ : ℝ} (E : Set α) (x : α) :
x ∈ thickening δ E ↔ ∃ z ∈ E, edist x z < ENNReal.ofReal δ :=
infEdist_lt_iff | theorem | Topology | [
"Mathlib.Topology.MetricSpace.HausdorffDistance"
] | Mathlib/Topology/MetricSpace/Thickening.lean | mem_thickening_iff_exists_edist_lt | null |
frontier_thickening_subset (E : Set α) {δ : ℝ} :
frontier (thickening δ E) ⊆ { x : α | infEdist x E = ENNReal.ofReal δ } :=
frontier_lt_subset_eq continuous_infEdist continuous_const
open scoped Function in -- required for scoped `on` notation | theorem | Topology | [
"Mathlib.Topology.MetricSpace.HausdorffDistance"
] | Mathlib/Topology/MetricSpace/Thickening.lean | frontier_thickening_subset | The frontier of the (open) thickening of a set is contained in an `EMetric.infEdist` level
set. |
frontier_thickening_disjoint (A : Set α) :
Pairwise (Disjoint on fun r : ℝ => frontier (thickening r A)) := by
refine (pairwise_disjoint_on _).2 fun r₁ r₂ hr => ?_
rcases le_total r₁ 0 with h₁ | h₁
· simp [thickening_of_nonpos h₁]
refine ((disjoint_singleton.2 fun h => hr.ne ?_).preimage _).mono (frontier_thickening_subset _)
(frontier_thickening_subset _)
apply_fun ENNReal.toReal at h
rwa [ENNReal.toReal_ofReal h₁, ENNReal.toReal_ofReal (h₁.trans hr.le)] at h | theorem | Topology | [
"Mathlib.Topology.MetricSpace.HausdorffDistance"
] | Mathlib/Topology/MetricSpace/Thickening.lean | frontier_thickening_disjoint | null |
subset_compl_thickening_compl_thickening_self (δ : ℝ) (E : Set α) :
E ⊆ (thickening δ (thickening δ E)ᶜ)ᶜ := by
intro x x_in_E
simp only [thickening, mem_compl_iff, mem_setOf_eq, not_lt]
apply EMetric.le_infEdist.mpr fun y hy ↦ ?_
simp only [mem_compl_iff, mem_setOf_eq, not_lt] at hy
simpa only [edist_comm] using le_trans hy <| EMetric.infEdist_le_edist_of_mem x_in_E | lemma | Topology | [
"Mathlib.Topology.MetricSpace.HausdorffDistance"
] | Mathlib/Topology/MetricSpace/Thickening.lean | subset_compl_thickening_compl_thickening_self | Any set is contained in the complement of the δ-thickening of the complement of its
δ-thickening. |
thickening_compl_thickening_self_subset_compl (δ : ℝ) (E : Set α) :
thickening δ (thickening δ E)ᶜ ⊆ Eᶜ := by
apply compl_subset_compl.mp
simpa only [compl_compl] using subset_compl_thickening_compl_thickening_self δ E
variable {X : Type u} [PseudoMetricSpace X] | lemma | Topology | [
"Mathlib.Topology.MetricSpace.HausdorffDistance"
] | Mathlib/Topology/MetricSpace/Thickening.lean | thickening_compl_thickening_self_subset_compl | The δ-thickening of the complement of the δ-thickening of a set is contained in the complement
of the set. |
mem_thickening_iff_infDist_lt {E : Set X} {x : X} (h : E.Nonempty) :
x ∈ thickening δ E ↔ infDist x E < δ :=
lt_ofReal_iff_toReal_lt (infEdist_ne_top h) | theorem | Topology | [
"Mathlib.Topology.MetricSpace.HausdorffDistance"
] | Mathlib/Topology/MetricSpace/Thickening.lean | mem_thickening_iff_infDist_lt | null |
mem_thickening_iff {E : Set X} {x : X} : x ∈ thickening δ E ↔ ∃ z ∈ E, dist x z < δ := by
have key_iff : ∀ z : X, edist x z < ENNReal.ofReal δ ↔ dist x z < δ := fun z ↦ by
rw [dist_edist, lt_ofReal_iff_toReal_lt (edist_ne_top _ _)]
simp_rw [mem_thickening_iff_exists_edist_lt, key_iff]
@[simp] | theorem | Topology | [
"Mathlib.Topology.MetricSpace.HausdorffDistance"
] | Mathlib/Topology/MetricSpace/Thickening.lean | mem_thickening_iff | A point in a metric space belongs to the (open) `δ`-thickening of a subset `E` if and only if
it is at distance less than `δ` from some point of `E`. |
thickening_singleton (δ : ℝ) (x : X) : thickening δ ({x} : Set X) = ball x δ := by
ext
simp [mem_thickening_iff] | theorem | Topology | [
"Mathlib.Topology.MetricSpace.HausdorffDistance"
] | Mathlib/Topology/MetricSpace/Thickening.lean | thickening_singleton | null |
ball_subset_thickening {x : X} {E : Set X} (hx : x ∈ E) (δ : ℝ) :
ball x δ ⊆ thickening δ E :=
Subset.trans (by simp) (thickening_subset_of_subset δ <| singleton_subset_iff.mpr hx) | theorem | Topology | [
"Mathlib.Topology.MetricSpace.HausdorffDistance"
] | Mathlib/Topology/MetricSpace/Thickening.lean | ball_subset_thickening | null |
thickening_eq_biUnion_ball {δ : ℝ} {E : Set X} : thickening δ E = ⋃ x ∈ E, ball x δ := by
ext x
simp only [mem_iUnion₂, exists_prop]
exact mem_thickening_iff | theorem | Topology | [
"Mathlib.Topology.MetricSpace.HausdorffDistance"
] | Mathlib/Topology/MetricSpace/Thickening.lean | thickening_eq_biUnion_ball | The (open) `δ`-thickening `Metric.thickening δ E` of a subset `E` in a metric space equals the
union of balls of radius `δ` centered at points of `E`. |
protected _root_.Bornology.IsBounded.thickening {δ : ℝ} {E : Set X} (h : IsBounded E) :
IsBounded (thickening δ E) := by
rcases E.eq_empty_or_nonempty with rfl | ⟨x, hx⟩
· simp
· refine (isBounded_iff_subset_closedBall x).2 ⟨δ + diam E, fun y hy ↦ ?_⟩
calc
dist y x ≤ infDist y E + diam E := dist_le_infDist_add_diam (x := y) h hx
_ ≤ δ + diam E := add_le_add_right ((mem_thickening_iff_infDist_lt ⟨x, hx⟩).1 hy).le _ | theorem | Topology | [
"Mathlib.Topology.MetricSpace.HausdorffDistance"
] | Mathlib/Topology/MetricSpace/Thickening.lean | _root_.Bornology.IsBounded.thickening | null |
cthickening (δ : ℝ) (E : Set α) : Set α :=
{ x : α | infEdist x E ≤ ENNReal.ofReal δ }
@[simp] | def | Topology | [
"Mathlib.Topology.MetricSpace.HausdorffDistance"
] | Mathlib/Topology/MetricSpace/Thickening.lean | cthickening | The closed `δ`-thickening `Metric.cthickening δ E` of a subset `E` in a pseudo emetric space
consists of those points that are at infimum distance at most `δ` from `E`. |
mem_cthickening_iff : x ∈ cthickening δ s ↔ infEdist x s ≤ ENNReal.ofReal δ :=
Iff.rfl | theorem | Topology | [
"Mathlib.Topology.MetricSpace.HausdorffDistance"
] | Mathlib/Topology/MetricSpace/Thickening.lean | mem_cthickening_iff | null |
eventually_notMem_cthickening_of_infEdist_pos {E : Set α} {x : α} (h : x ∉ closure E) :
∀ᶠ δ in 𝓝 (0 : ℝ), x ∉ Metric.cthickening δ E := by
obtain ⟨ε, ⟨ε_pos, ε_lt⟩⟩ := exists_real_pos_lt_infEdist_of_notMem_closure h
filter_upwards [eventually_lt_nhds ε_pos] with δ hδ
simp only [cthickening, mem_setOf_eq, not_le]
exact ((ofReal_lt_ofReal_iff ε_pos).mpr hδ).trans ε_lt
@[deprecated (since := "2025-05-23")]
alias eventually_not_mem_cthickening_of_infEdist_pos :=
eventually_notMem_cthickening_of_infEdist_pos | lemma | Topology | [
"Mathlib.Topology.MetricSpace.HausdorffDistance"
] | Mathlib/Topology/MetricSpace/Thickening.lean | eventually_notMem_cthickening_of_infEdist_pos | An exterior point of a subset `E` (i.e., a point outside the closure of `E`) is not in the
closed `δ`-thickening of `E` for small enough positive `δ`. |
mem_cthickening_of_edist_le (x y : α) (δ : ℝ) (E : Set α) (h : y ∈ E)
(h' : edist x y ≤ ENNReal.ofReal δ) : x ∈ cthickening δ E :=
(infEdist_le_edist_of_mem h).trans h' | theorem | Topology | [
"Mathlib.Topology.MetricSpace.HausdorffDistance"
] | Mathlib/Topology/MetricSpace/Thickening.lean | mem_cthickening_of_edist_le | null |
mem_cthickening_of_dist_le {α : Type*} [PseudoMetricSpace α] (x y : α) (δ : ℝ) (E : Set α)
(h : y ∈ E) (h' : dist x y ≤ δ) : x ∈ cthickening δ E := by
apply mem_cthickening_of_edist_le x y δ E h
rw [edist_dist]
exact ENNReal.ofReal_le_ofReal h' | theorem | Topology | [
"Mathlib.Topology.MetricSpace.HausdorffDistance"
] | Mathlib/Topology/MetricSpace/Thickening.lean | mem_cthickening_of_dist_le | null |
cthickening_eq_preimage_infEdist (δ : ℝ) (E : Set α) :
cthickening δ E = (fun x => infEdist x E) ⁻¹' Iic (ENNReal.ofReal δ) :=
rfl | theorem | Topology | [
"Mathlib.Topology.MetricSpace.HausdorffDistance"
] | Mathlib/Topology/MetricSpace/Thickening.lean | cthickening_eq_preimage_infEdist | null |
isClosed_cthickening {δ : ℝ} {E : Set α} : IsClosed (cthickening δ E) :=
IsClosed.preimage continuous_infEdist isClosed_Iic | theorem | Topology | [
"Mathlib.Topology.MetricSpace.HausdorffDistance"
] | Mathlib/Topology/MetricSpace/Thickening.lean | isClosed_cthickening | The closed thickening is a closed set. |
@[simp]
cthickening_empty (δ : ℝ) : cthickening δ (∅ : Set α) = ∅ := by
simp only [cthickening, ENNReal.ofReal_ne_top, setOf_false, infEdist_empty, top_le_iff] | theorem | Topology | [
"Mathlib.Topology.MetricSpace.HausdorffDistance"
] | Mathlib/Topology/MetricSpace/Thickening.lean | cthickening_empty | The closed thickening of the empty set is empty. |
cthickening_of_nonpos {δ : ℝ} (hδ : δ ≤ 0) (E : Set α) : cthickening δ E = closure E := by
ext x
simp [mem_closure_iff_infEdist_zero, cthickening, ENNReal.ofReal_eq_zero.2 hδ] | theorem | Topology | [
"Mathlib.Topology.MetricSpace.HausdorffDistance"
] | Mathlib/Topology/MetricSpace/Thickening.lean | cthickening_of_nonpos | null |
@[simp]
cthickening_zero (E : Set α) : cthickening 0 E = closure E :=
cthickening_of_nonpos le_rfl E | theorem | Topology | [
"Mathlib.Topology.MetricSpace.HausdorffDistance"
] | Mathlib/Topology/MetricSpace/Thickening.lean | cthickening_zero | The closed thickening with radius zero is the closure of the set. |
cthickening_max_zero (δ : ℝ) (E : Set α) : cthickening (max 0 δ) E = cthickening δ E := by
cases le_total δ 0 <;> simp [cthickening_of_nonpos, *] | theorem | Topology | [
"Mathlib.Topology.MetricSpace.HausdorffDistance"
] | Mathlib/Topology/MetricSpace/Thickening.lean | cthickening_max_zero | null |
cthickening_mono {δ₁ δ₂ : ℝ} (hle : δ₁ ≤ δ₂) (E : Set α) :
cthickening δ₁ E ⊆ cthickening δ₂ E :=
preimage_mono (Iic_subset_Iic.mpr (ENNReal.ofReal_le_ofReal hle))
@[simp] | theorem | Topology | [
"Mathlib.Topology.MetricSpace.HausdorffDistance"
] | Mathlib/Topology/MetricSpace/Thickening.lean | cthickening_mono | The closed thickening `Metric.cthickening δ E` of a fixed subset `E` is an increasing function
of the thickening radius `δ`. |
cthickening_singleton {α : Type*} [PseudoMetricSpace α] (x : α) {δ : ℝ} (hδ : 0 ≤ δ) :
cthickening δ ({x} : Set α) = closedBall x δ := by
ext y
simp [cthickening, edist_dist, ENNReal.ofReal_le_ofReal_iff hδ] | theorem | Topology | [
"Mathlib.Topology.MetricSpace.HausdorffDistance"
] | Mathlib/Topology/MetricSpace/Thickening.lean | cthickening_singleton | null |
closedBall_subset_cthickening_singleton {α : Type*} [PseudoMetricSpace α] (x : α) (δ : ℝ) :
closedBall x δ ⊆ cthickening δ ({x} : Set α) := by
rcases lt_or_ge δ 0 with (hδ | hδ)
· simp only [closedBall_eq_empty.mpr hδ, empty_subset]
· simp only [cthickening_singleton x hδ, Subset.rfl] | theorem | Topology | [
"Mathlib.Topology.MetricSpace.HausdorffDistance"
] | Mathlib/Topology/MetricSpace/Thickening.lean | closedBall_subset_cthickening_singleton | null |
cthickening_subset_of_subset (δ : ℝ) {E₁ E₂ : Set α} (h : E₁ ⊆ E₂) :
cthickening δ E₁ ⊆ cthickening δ E₂ := fun _ hx => le_trans (infEdist_anti h) hx | theorem | Topology | [
"Mathlib.Topology.MetricSpace.HausdorffDistance"
] | Mathlib/Topology/MetricSpace/Thickening.lean | cthickening_subset_of_subset | The closed thickening `Metric.cthickening δ E` with a fixed thickening radius `δ` is
an increasing function of the subset `E`. |
cthickening_subset_thickening {δ₁ : ℝ≥0} {δ₂ : ℝ} (hlt : (δ₁ : ℝ) < δ₂) (E : Set α) :
cthickening δ₁ E ⊆ thickening δ₂ E := fun _ hx =>
hx.out.trans_lt ((ENNReal.ofReal_lt_ofReal_iff (lt_of_le_of_lt δ₁.prop hlt)).mpr hlt) | theorem | Topology | [
"Mathlib.Topology.MetricSpace.HausdorffDistance"
] | Mathlib/Topology/MetricSpace/Thickening.lean | cthickening_subset_thickening | null |
cthickening_subset_thickening' {δ₁ δ₂ : ℝ} (δ₂_pos : 0 < δ₂) (hlt : δ₁ < δ₂) (E : Set α) :
cthickening δ₁ E ⊆ thickening δ₂ E := fun _ hx =>
lt_of_le_of_lt hx.out ((ENNReal.ofReal_lt_ofReal_iff δ₂_pos).mpr hlt) | theorem | Topology | [
"Mathlib.Topology.MetricSpace.HausdorffDistance"
] | Mathlib/Topology/MetricSpace/Thickening.lean | cthickening_subset_thickening' | The closed thickening `Metric.cthickening δ₁ E` is contained in the open thickening
`Metric.thickening δ₂ E` if the radius of the latter is positive and larger. |
thickening_subset_cthickening (δ : ℝ) (E : Set α) : thickening δ E ⊆ cthickening δ E := by
intro x hx
rw [thickening, mem_setOf_eq] at hx
exact hx.le | theorem | Topology | [
"Mathlib.Topology.MetricSpace.HausdorffDistance"
] | Mathlib/Topology/MetricSpace/Thickening.lean | thickening_subset_cthickening | The open thickening `Metric.thickening δ E` is contained in the closed thickening
`Metric.cthickening δ E` with the same radius. |
thickening_subset_cthickening_of_le {δ₁ δ₂ : ℝ} (hle : δ₁ ≤ δ₂) (E : Set α) :
thickening δ₁ E ⊆ cthickening δ₂ E :=
(thickening_subset_cthickening δ₁ E).trans (cthickening_mono hle E) | theorem | Topology | [
"Mathlib.Topology.MetricSpace.HausdorffDistance"
] | Mathlib/Topology/MetricSpace/Thickening.lean | thickening_subset_cthickening_of_le | null |
_root_.Bornology.IsBounded.cthickening {α : Type*} [PseudoMetricSpace α] {δ : ℝ} {E : Set α}
(h : IsBounded E) : IsBounded (cthickening δ E) := by
have : IsBounded (thickening (max (δ + 1) 1) E) := h.thickening
apply this.subset
exact cthickening_subset_thickening' (zero_lt_one.trans_le (le_max_right _ _))
((lt_add_one _).trans_le (le_max_left _ _)) _ | theorem | Topology | [
"Mathlib.Topology.MetricSpace.HausdorffDistance"
] | Mathlib/Topology/MetricSpace/Thickening.lean | _root_.Bornology.IsBounded.cthickening | null |
protected _root_.IsCompact.cthickening
{α : Type*} [PseudoMetricSpace α] [ProperSpace α] {s : Set α}
(hs : IsCompact s) {r : ℝ} : IsCompact (cthickening r s) :=
isCompact_of_isClosed_isBounded isClosed_cthickening hs.isBounded.cthickening | theorem | Topology | [
"Mathlib.Topology.MetricSpace.HausdorffDistance"
] | Mathlib/Topology/MetricSpace/Thickening.lean | _root_.IsCompact.cthickening | null |
thickening_subset_interior_cthickening (δ : ℝ) (E : Set α) :
thickening δ E ⊆ interior (cthickening δ E) :=
(subset_interior_iff_isOpen.mpr isOpen_thickening).trans
(interior_mono (thickening_subset_cthickening δ E)) | theorem | Topology | [
"Mathlib.Topology.MetricSpace.HausdorffDistance"
] | Mathlib/Topology/MetricSpace/Thickening.lean | thickening_subset_interior_cthickening | null |
closure_thickening_subset_cthickening (δ : ℝ) (E : Set α) :
closure (thickening δ E) ⊆ cthickening δ E :=
(closure_mono (thickening_subset_cthickening δ E)).trans isClosed_cthickening.closure_subset | theorem | Topology | [
"Mathlib.Topology.MetricSpace.HausdorffDistance"
] | Mathlib/Topology/MetricSpace/Thickening.lean | closure_thickening_subset_cthickening | null |
closure_subset_cthickening (δ : ℝ) (E : Set α) : closure E ⊆ cthickening δ E := by
rw [← cthickening_of_nonpos (min_le_right δ 0)]
exact cthickening_mono (min_le_left δ 0) E | theorem | Topology | [
"Mathlib.Topology.MetricSpace.HausdorffDistance"
] | Mathlib/Topology/MetricSpace/Thickening.lean | closure_subset_cthickening | The closed thickening of a set contains the closure of the set. |
closure_subset_thickening {δ : ℝ} (δ_pos : 0 < δ) (E : Set α) :
closure E ⊆ thickening δ E := by
rw [← cthickening_zero]
exact cthickening_subset_thickening' δ_pos δ_pos E | theorem | Topology | [
"Mathlib.Topology.MetricSpace.HausdorffDistance"
] | Mathlib/Topology/MetricSpace/Thickening.lean | closure_subset_thickening | The (open) thickening of a set contains the closure of the set. |
self_subset_thickening {δ : ℝ} (δ_pos : 0 < δ) (E : Set α) : E ⊆ thickening δ E :=
(@subset_closure _ _ E).trans (closure_subset_thickening δ_pos E) | theorem | Topology | [
"Mathlib.Topology.MetricSpace.HausdorffDistance"
] | Mathlib/Topology/MetricSpace/Thickening.lean | self_subset_thickening | A set is contained in its own (open) thickening. |
self_subset_cthickening {δ : ℝ} (E : Set α) : E ⊆ cthickening δ E :=
subset_closure.trans (closure_subset_cthickening δ E) | theorem | Topology | [
"Mathlib.Topology.MetricSpace.HausdorffDistance"
] | Mathlib/Topology/MetricSpace/Thickening.lean | self_subset_cthickening | A set is contained in its own closed thickening. |
thickening_mem_nhdsSet (E : Set α) {δ : ℝ} (hδ : 0 < δ) : thickening δ E ∈ 𝓝ˢ E :=
isOpen_thickening.mem_nhdsSet.2 <| self_subset_thickening hδ E | theorem | Topology | [
"Mathlib.Topology.MetricSpace.HausdorffDistance"
] | Mathlib/Topology/MetricSpace/Thickening.lean | thickening_mem_nhdsSet | null |
cthickening_mem_nhdsSet (E : Set α) {δ : ℝ} (hδ : 0 < δ) : cthickening δ E ∈ 𝓝ˢ E :=
mem_of_superset (thickening_mem_nhdsSet E hδ) (thickening_subset_cthickening _ _)
@[simp] | theorem | Topology | [
"Mathlib.Topology.MetricSpace.HausdorffDistance"
] | Mathlib/Topology/MetricSpace/Thickening.lean | cthickening_mem_nhdsSet | null |
thickening_union (δ : ℝ) (s t : Set α) :
thickening δ (s ∪ t) = thickening δ s ∪ thickening δ t := by
simp_rw [thickening, infEdist_union, min_lt_iff, setOf_or]
@[simp] | theorem | Topology | [
"Mathlib.Topology.MetricSpace.HausdorffDistance"
] | Mathlib/Topology/MetricSpace/Thickening.lean | thickening_union | null |
cthickening_union (δ : ℝ) (s t : Set α) :
cthickening δ (s ∪ t) = cthickening δ s ∪ cthickening δ t := by
simp_rw [cthickening, infEdist_union, min_le_iff, setOf_or]
@[simp] | theorem | Topology | [
"Mathlib.Topology.MetricSpace.HausdorffDistance"
] | Mathlib/Topology/MetricSpace/Thickening.lean | cthickening_union | null |
thickening_iUnion (δ : ℝ) (f : ι → Set α) :
thickening δ (⋃ i, f i) = ⋃ i, thickening δ (f i) := by
simp_rw [thickening, infEdist_iUnion, iInf_lt_iff, setOf_exists] | theorem | Topology | [
"Mathlib.Topology.MetricSpace.HausdorffDistance"
] | Mathlib/Topology/MetricSpace/Thickening.lean | thickening_iUnion | null |
thickening_biUnion {ι : Type*} (δ : ℝ) (f : ι → Set α) (I : Set ι) :
thickening δ (⋃ i ∈ I, f i) = ⋃ i ∈ I, thickening δ (f i) := by simp only [thickening_iUnion] | lemma | Topology | [
"Mathlib.Topology.MetricSpace.HausdorffDistance"
] | Mathlib/Topology/MetricSpace/Thickening.lean | thickening_biUnion | null |
ediam_cthickening_le (ε : ℝ≥0) :
EMetric.diam (cthickening ε s) ≤ EMetric.diam s + 2 * ε := by
refine diam_le fun x hx y hy => ENNReal.le_of_forall_pos_le_add fun δ hδ _ => ?_
rw [mem_cthickening_iff, ENNReal.ofReal_coe_nnreal] at hx hy
have hε : (ε : ℝ≥0∞) < ε + δ := ENNReal.coe_lt_coe.2 (lt_add_of_pos_right _ hδ)
replace hx := hx.trans_lt hε
obtain ⟨x', hx', hxx'⟩ := infEdist_lt_iff.mp hx
calc
edist x y ≤ edist x x' + edist y x' := edist_triangle_right _ _ _
_ ≤ ε + δ + (infEdist y s + EMetric.diam s) :=
add_le_add hxx'.le (edist_le_infEdist_add_ediam hx')
_ ≤ ε + δ + (ε + EMetric.diam s) := by grw [hy]
_ = _ := by rw [two_mul]; ac_rfl | theorem | Topology | [
"Mathlib.Topology.MetricSpace.HausdorffDistance"
] | Mathlib/Topology/MetricSpace/Thickening.lean | ediam_cthickening_le | null |
ediam_thickening_le (ε : ℝ≥0) : EMetric.diam (thickening ε s) ≤ EMetric.diam s + 2 * ε :=
(EMetric.diam_mono <| thickening_subset_cthickening _ _).trans <| ediam_cthickening_le _ | theorem | Topology | [
"Mathlib.Topology.MetricSpace.HausdorffDistance"
] | Mathlib/Topology/MetricSpace/Thickening.lean | ediam_thickening_le | null |
diam_cthickening_le {α : Type*} [PseudoMetricSpace α] (s : Set α) (hε : 0 ≤ ε) :
diam (cthickening ε s) ≤ diam s + 2 * ε := by
lift ε to ℝ≥0 using hε
refine (toReal_le_add' (ediam_cthickening_le _) ?_ ?_).trans_eq ?_
· exact fun h ↦ top_unique <| h ▸ EMetric.diam_mono (self_subset_cthickening _)
· simp [mul_eq_top]
· simp [diam] | theorem | Topology | [
"Mathlib.Topology.MetricSpace.HausdorffDistance"
] | Mathlib/Topology/MetricSpace/Thickening.lean | diam_cthickening_le | null |
diam_thickening_le {α : Type*} [PseudoMetricSpace α] (s : Set α) (hε : 0 ≤ ε) :
diam (thickening ε s) ≤ diam s + 2 * ε := by
by_cases hs : IsBounded s
· exact (diam_mono (thickening_subset_cthickening _ _) hs.cthickening).trans
(diam_cthickening_le _ hε)
obtain rfl | hε := hε.eq_or_lt
· simp [thickening_of_nonpos, diam_nonneg]
· rw [diam_eq_zero_of_unbounded (mt (IsBounded.subset · <| self_subset_thickening hε _) hs)]
positivity
@[simp] | theorem | Topology | [
"Mathlib.Topology.MetricSpace.HausdorffDistance"
] | Mathlib/Topology/MetricSpace/Thickening.lean | diam_thickening_le | null |
thickening_closure : thickening δ (closure s) = thickening δ s := by
simp_rw [thickening, infEdist_closure]
@[simp] | theorem | Topology | [
"Mathlib.Topology.MetricSpace.HausdorffDistance"
] | Mathlib/Topology/MetricSpace/Thickening.lean | thickening_closure | null |
cthickening_closure : cthickening δ (closure s) = cthickening δ s := by
simp_rw [cthickening, infEdist_closure] | theorem | Topology | [
"Mathlib.Topology.MetricSpace.HausdorffDistance"
] | Mathlib/Topology/MetricSpace/Thickening.lean | cthickening_closure | null |
thickening_eq_empty_iff_of_pos (hε : 0 < ε) :
thickening ε s = ∅ ↔ s = ∅ :=
⟨fun h ↦ subset_eq_empty (self_subset_thickening hε _) h, by simp +contextual⟩ | lemma | Topology | [
"Mathlib.Topology.MetricSpace.HausdorffDistance"
] | Mathlib/Topology/MetricSpace/Thickening.lean | thickening_eq_empty_iff_of_pos | null |
thickening_nonempty_iff_of_pos (hε : 0 < ε) :
(thickening ε s).Nonempty ↔ s.Nonempty := by
simp [nonempty_iff_ne_empty, thickening_eq_empty_iff_of_pos hε]
@[simp] lemma thickening_eq_empty_iff : thickening ε s = ∅ ↔ ε ≤ 0 ∨ s = ∅ := by
obtain hε | hε := lt_or_ge 0 ε
· simp [thickening_eq_empty_iff_of_pos, hε]
· simp [hε, thickening_of_nonpos hε]
@[simp] lemma thickening_nonempty_iff : (thickening ε s).Nonempty ↔ 0 < ε ∧ s.Nonempty := by
simp [nonempty_iff_ne_empty]
open ENNReal | lemma | Topology | [
"Mathlib.Topology.MetricSpace.HausdorffDistance"
] | Mathlib/Topology/MetricSpace/Thickening.lean | thickening_nonempty_iff_of_pos | null |
_root_.Disjoint.exists_thickenings (hst : Disjoint s t) (hs : IsCompact s)
(ht : IsClosed t) :
∃ δ, 0 < δ ∧ Disjoint (thickening δ s) (thickening δ t) := by
obtain ⟨r, hr, h⟩ := exists_pos_forall_lt_edist hs ht hst
refine ⟨r / 2, half_pos (NNReal.coe_pos.2 hr), ?_⟩
rw [disjoint_iff_inf_le]
rintro z ⟨hzs, hzt⟩
rw [mem_thickening_iff_exists_edist_lt] at hzs hzt
rw [← NNReal.coe_two, ← NNReal.coe_div, ENNReal.ofReal_coe_nnreal] at hzs hzt
obtain ⟨x, hx, hzx⟩ := hzs
obtain ⟨y, hy, hzy⟩ := hzt
refine (h x hx y hy).not_ge ?_
calc
edist x y ≤ edist z x + edist z y := edist_triangle_left _ _ _
_ ≤ ↑(r / 2) + ↑(r / 2) := add_le_add hzx.le hzy.le
_ = r := by rw [← ENNReal.coe_add, add_halves] | theorem | Topology | [
"Mathlib.Topology.MetricSpace.HausdorffDistance"
] | Mathlib/Topology/MetricSpace/Thickening.lean | _root_.Disjoint.exists_thickenings | null |
_root_.Disjoint.exists_cthickenings (hst : Disjoint s t) (hs : IsCompact s)
(ht : IsClosed t) :
∃ δ, 0 < δ ∧ Disjoint (cthickening δ s) (cthickening δ t) := by
obtain ⟨δ, hδ, h⟩ := hst.exists_thickenings hs ht
refine ⟨δ / 2, half_pos hδ, h.mono ?_ ?_⟩ <;>
exact cthickening_subset_thickening' hδ (half_lt_self hδ) _ | theorem | Topology | [
"Mathlib.Topology.MetricSpace.HausdorffDistance"
] | Mathlib/Topology/MetricSpace/Thickening.lean | _root_.Disjoint.exists_cthickenings | null |
_root_.IsCompact.exists_cthickening_subset_open (hs : IsCompact s) (ht : IsOpen t)
(hst : s ⊆ t) :
∃ δ, 0 < δ ∧ cthickening δ s ⊆ t :=
(hst.disjoint_compl_right.exists_cthickenings hs ht.isClosed_compl).imp fun _ h =>
⟨h.1, disjoint_compl_right_iff_subset.1 <| h.2.mono_right <| self_subset_cthickening _⟩ | theorem | Topology | [
"Mathlib.Topology.MetricSpace.HausdorffDistance"
] | Mathlib/Topology/MetricSpace/Thickening.lean | _root_.IsCompact.exists_cthickening_subset_open | If `s` is compact, `t` is open and `s ⊆ t`, some `cthickening` of `s` is contained in `t`. |
_root_.IsCompact.exists_isCompact_cthickening [LocallyCompactSpace α] (hs : IsCompact s) :
∃ δ, 0 < δ ∧ IsCompact (cthickening δ s) := by
rcases exists_compact_superset hs with ⟨K, K_compact, hK⟩
rcases hs.exists_cthickening_subset_open isOpen_interior hK with ⟨δ, δpos, hδ⟩
refine ⟨δ, δpos, ?_⟩
exact K_compact.of_isClosed_subset isClosed_cthickening (hδ.trans interior_subset) | theorem | Topology | [
"Mathlib.Topology.MetricSpace.HausdorffDistance"
] | Mathlib/Topology/MetricSpace/Thickening.lean | _root_.IsCompact.exists_isCompact_cthickening | null |
_root_.IsCompact.exists_thickening_subset_open (hs : IsCompact s) (ht : IsOpen t)
(hst : s ⊆ t) : ∃ δ, 0 < δ ∧ thickening δ s ⊆ t :=
let ⟨δ, h₀, hδ⟩ := hs.exists_cthickening_subset_open ht hst
⟨δ, h₀, (thickening_subset_cthickening _ _).trans hδ⟩ | theorem | Topology | [
"Mathlib.Topology.MetricSpace.HausdorffDistance"
] | Mathlib/Topology/MetricSpace/Thickening.lean | _root_.IsCompact.exists_thickening_subset_open | null |
hasBasis_nhdsSet_thickening {K : Set α} (hK : IsCompact K) :
(𝓝ˢ K).HasBasis (fun δ : ℝ => 0 < δ) fun δ => thickening δ K :=
(hasBasis_nhdsSet K).to_hasBasis' (fun _U hU => hK.exists_thickening_subset_open hU.1 hU.2)
fun _ => thickening_mem_nhdsSet K | theorem | Topology | [
"Mathlib.Topology.MetricSpace.HausdorffDistance"
] | Mathlib/Topology/MetricSpace/Thickening.lean | hasBasis_nhdsSet_thickening | null |
hasBasis_nhdsSet_cthickening {K : Set α} (hK : IsCompact K) :
(𝓝ˢ K).HasBasis (fun δ : ℝ => 0 < δ) fun δ => cthickening δ K :=
(hasBasis_nhdsSet K).to_hasBasis' (fun _U hU => hK.exists_cthickening_subset_open hU.1 hU.2)
fun _ => cthickening_mem_nhdsSet K | theorem | Topology | [
"Mathlib.Topology.MetricSpace.HausdorffDistance"
] | Mathlib/Topology/MetricSpace/Thickening.lean | hasBasis_nhdsSet_cthickening | null |
cthickening_eq_iInter_cthickening' {δ : ℝ} (s : Set ℝ) (hsδ : s ⊆ Ioi δ)
(hs : ∀ ε, δ < ε → (s ∩ Ioc δ ε).Nonempty) (E : Set α) :
cthickening δ E = ⋂ ε ∈ s, cthickening ε E := by
apply Subset.antisymm
· exact subset_iInter₂ fun _ hε => cthickening_mono (le_of_lt (hsδ hε)) E
· unfold cthickening
intro x hx
simp only [mem_iInter, mem_setOf_eq] at *
apply ENNReal.le_of_forall_pos_le_add
intro η η_pos _
rcases hs (δ + η) (lt_add_of_pos_right _ (NNReal.coe_pos.mpr η_pos)) with ⟨ε, ⟨hsε, hε⟩⟩
apply ((hx ε hsε).trans (ENNReal.ofReal_le_ofReal hε.2)).trans
rw [ENNReal.coe_nnreal_eq η]
exact ENNReal.ofReal_add_le | theorem | Topology | [
"Mathlib.Topology.MetricSpace.HausdorffDistance"
] | Mathlib/Topology/MetricSpace/Thickening.lean | cthickening_eq_iInter_cthickening' | null |
cthickening_eq_iInter_cthickening {δ : ℝ} (E : Set α) :
cthickening δ E = ⋂ (ε : ℝ) (_ : δ < ε), cthickening ε E := by
apply cthickening_eq_iInter_cthickening' (Ioi δ) rfl.subset
simp_rw [inter_eq_right.mpr Ioc_subset_Ioi_self]
exact fun _ hε => nonempty_Ioc.mpr hε | theorem | Topology | [
"Mathlib.Topology.MetricSpace.HausdorffDistance"
] | Mathlib/Topology/MetricSpace/Thickening.lean | cthickening_eq_iInter_cthickening | null |
cthickening_eq_iInter_thickening' {δ : ℝ} (δ_nn : 0 ≤ δ) (s : Set ℝ) (hsδ : s ⊆ Ioi δ)
(hs : ∀ ε, δ < ε → (s ∩ Ioc δ ε).Nonempty) (E : Set α) :
cthickening δ E = ⋂ ε ∈ s, thickening ε E := by
refine (subset_iInter₂ fun ε hε => ?_).antisymm ?_
· obtain ⟨ε', -, hε'⟩ := hs ε (hsδ hε)
have ss := cthickening_subset_thickening' (lt_of_le_of_lt δ_nn hε'.1) hε'.1 E
exact ss.trans (thickening_mono hε'.2 E)
· rw [cthickening_eq_iInter_cthickening' s hsδ hs E]
exact iInter₂_mono fun ε _ => thickening_subset_cthickening ε E | theorem | Topology | [
"Mathlib.Topology.MetricSpace.HausdorffDistance"
] | Mathlib/Topology/MetricSpace/Thickening.lean | cthickening_eq_iInter_thickening' | null |
cthickening_eq_iInter_thickening {δ : ℝ} (δ_nn : 0 ≤ δ) (E : Set α) :
cthickening δ E = ⋂ (ε : ℝ) (_ : δ < ε), thickening ε E := by
apply cthickening_eq_iInter_thickening' δ_nn (Ioi δ) rfl.subset
simp_rw [inter_eq_right.mpr Ioc_subset_Ioi_self]
exact fun _ hε => nonempty_Ioc.mpr hε | theorem | Topology | [
"Mathlib.Topology.MetricSpace.HausdorffDistance"
] | Mathlib/Topology/MetricSpace/Thickening.lean | cthickening_eq_iInter_thickening | null |
cthickening_eq_iInter_thickening'' (δ : ℝ) (E : Set α) :
cthickening δ E = ⋂ (ε : ℝ) (_ : max 0 δ < ε), thickening ε E := by
rw [← cthickening_max_zero, cthickening_eq_iInter_thickening]
exact le_max_left _ _ | theorem | Topology | [
"Mathlib.Topology.MetricSpace.HausdorffDistance"
] | Mathlib/Topology/MetricSpace/Thickening.lean | cthickening_eq_iInter_thickening'' | null |
closure_eq_iInter_cthickening' (E : Set α) (s : Set ℝ)
(hs : ∀ ε, 0 < ε → (s ∩ Ioc 0 ε).Nonempty) : closure E = ⋂ δ ∈ s, cthickening δ E := by
by_cases hs₀ : s ⊆ Ioi 0
· rw [← cthickening_zero]
apply cthickening_eq_iInter_cthickening' _ hs₀ hs
obtain ⟨δ, hδs, δ_nonpos⟩ := not_subset.mp hs₀
rw [Set.mem_Ioi, not_lt] at δ_nonpos
apply Subset.antisymm
· exact subset_iInter₂ fun ε _ => closure_subset_cthickening ε E
· rw [← cthickening_of_nonpos δ_nonpos E]
exact biInter_subset_of_mem hδs | theorem | Topology | [
"Mathlib.Topology.MetricSpace.HausdorffDistance"
] | Mathlib/Topology/MetricSpace/Thickening.lean | closure_eq_iInter_cthickening' | The closure of a set equals the intersection of its closed thickenings of positive radii
accumulating at zero. |
closure_eq_iInter_cthickening (E : Set α) :
closure E = ⋂ (δ : ℝ) (_ : 0 < δ), cthickening δ E := by
rw [← cthickening_zero]
exact cthickening_eq_iInter_cthickening E | theorem | Topology | [
"Mathlib.Topology.MetricSpace.HausdorffDistance"
] | Mathlib/Topology/MetricSpace/Thickening.lean | closure_eq_iInter_cthickening | The closure of a set equals the intersection of its closed thickenings of positive radii. |
closure_eq_iInter_thickening' (E : Set α) (s : Set ℝ) (hs₀ : s ⊆ Ioi 0)
(hs : ∀ ε, 0 < ε → (s ∩ Ioc 0 ε).Nonempty) : closure E = ⋂ δ ∈ s, thickening δ E := by
rw [← cthickening_zero]
apply cthickening_eq_iInter_thickening' le_rfl _ hs₀ hs | theorem | Topology | [
"Mathlib.Topology.MetricSpace.HausdorffDistance"
] | Mathlib/Topology/MetricSpace/Thickening.lean | closure_eq_iInter_thickening' | The closure of a set equals the intersection of its open thickenings of positive radii
accumulating at zero. |
closure_eq_iInter_thickening (E : Set α) :
closure E = ⋂ (δ : ℝ) (_ : 0 < δ), thickening δ E := by
rw [← cthickening_zero]
exact cthickening_eq_iInter_thickening rfl.ge E | theorem | Topology | [
"Mathlib.Topology.MetricSpace.HausdorffDistance"
] | Mathlib/Topology/MetricSpace/Thickening.lean | closure_eq_iInter_thickening | The closure of a set equals the intersection of its (open) thickenings of positive radii. |
frontier_cthickening_subset (E : Set α) {δ : ℝ} :
frontier (cthickening δ E) ⊆ { x : α | infEdist x E = ENNReal.ofReal δ } :=
frontier_le_subset_eq continuous_infEdist continuous_const | theorem | Topology | [
"Mathlib.Topology.MetricSpace.HausdorffDistance"
] | Mathlib/Topology/MetricSpace/Thickening.lean | frontier_cthickening_subset | The frontier of the closed thickening of a set is contained in an `EMetric.infEdist` level
set. |
closedBall_subset_cthickening {α : Type*} [PseudoMetricSpace α] {x : α} {E : Set α}
(hx : x ∈ E) (δ : ℝ) : closedBall x δ ⊆ cthickening δ E := by
refine (closedBall_subset_cthickening_singleton _ _).trans (cthickening_subset_of_subset _ ?_)
simpa using hx | theorem | Topology | [
"Mathlib.Topology.MetricSpace.HausdorffDistance"
] | Mathlib/Topology/MetricSpace/Thickening.lean | closedBall_subset_cthickening | The closed ball of radius `δ` centered at a point of `E` is included in the closed
thickening of `E`. |
cthickening_subset_iUnion_closedBall_of_lt {α : Type*} [PseudoMetricSpace α] (E : Set α)
{δ δ' : ℝ} (hδ₀ : 0 < δ') (hδδ' : δ < δ') : cthickening δ E ⊆ ⋃ x ∈ E, closedBall x δ' := by
refine (cthickening_subset_thickening' hδ₀ hδδ' E).trans fun x hx => ?_
obtain ⟨y, hy₁, hy₂⟩ := mem_thickening_iff.mp hx
exact mem_iUnion₂.mpr ⟨y, hy₁, hy₂.le⟩ | theorem | Topology | [
"Mathlib.Topology.MetricSpace.HausdorffDistance"
] | Mathlib/Topology/MetricSpace/Thickening.lean | cthickening_subset_iUnion_closedBall_of_lt | null |
_root_.IsCompact.cthickening_eq_biUnion_closedBall {α : Type*} [PseudoMetricSpace α]
{δ : ℝ} {E : Set α} (hE : IsCompact E) (hδ : 0 ≤ δ) :
cthickening δ E = ⋃ x ∈ E, closedBall x δ := by
rcases eq_empty_or_nonempty E with (rfl | hne)
· simp only [cthickening_empty, biUnion_empty]
refine Subset.antisymm (fun x hx ↦ ?_)
(iUnion₂_subset fun x hx ↦ closedBall_subset_cthickening hx _)
obtain ⟨y, yE, hy⟩ : ∃ y ∈ E, infEdist x E = edist x y := hE.exists_infEdist_eq_edist hne _
have D1 : edist x y ≤ ENNReal.ofReal δ := (le_of_eq hy.symm).trans hx
have D2 : dist x y ≤ δ := by
rw [edist_dist] at D1
exact (ENNReal.ofReal_le_ofReal_iff hδ).1 D1
exact mem_biUnion yE D2 | theorem | Topology | [
"Mathlib.Topology.MetricSpace.HausdorffDistance"
] | Mathlib/Topology/MetricSpace/Thickening.lean | _root_.IsCompact.cthickening_eq_biUnion_closedBall | The closed thickening of a compact set `E` is the union of the balls `Metric.closedBall x δ`
over `x ∈ E`.
See also `Metric.cthickening_eq_biUnion_closedBall`. |
cthickening_eq_biUnion_closedBall {α : Type*} [PseudoMetricSpace α] [ProperSpace α]
(E : Set α) (hδ : 0 ≤ δ) : cthickening δ E = ⋃ x ∈ closure E, closedBall x δ := by
rcases eq_empty_or_nonempty E with (rfl | hne)
· simp only [cthickening_empty, biUnion_empty, closure_empty]
rw [← cthickening_closure]
refine Subset.antisymm (fun x hx ↦ ?_)
(iUnion₂_subset fun x hx ↦ closedBall_subset_cthickening hx _)
obtain ⟨y, yE, hy⟩ : ∃ y ∈ closure E, infDist x (closure E) = dist x y :=
isClosed_closure.exists_infDist_eq_dist (closure_nonempty_iff.mpr hne) x
replace hy : dist x y ≤ δ :=
(ENNReal.ofReal_le_ofReal_iff hδ).mp
(((congr_arg ENNReal.ofReal hy.symm).le.trans ENNReal.ofReal_toReal_le).trans hx)
exact mem_biUnion yE hy
nonrec theorem _root_.IsClosed.cthickening_eq_biUnion_closedBall {α : Type*} [PseudoMetricSpace α]
[ProperSpace α] {E : Set α} (hE : IsClosed E) (hδ : 0 ≤ δ) :
cthickening δ E = ⋃ x ∈ E, closedBall x δ := by
rw [cthickening_eq_biUnion_closedBall E hδ, hE.closure_eq] | theorem | Topology | [
"Mathlib.Topology.MetricSpace.HausdorffDistance"
] | Mathlib/Topology/MetricSpace/Thickening.lean | cthickening_eq_biUnion_closedBall | null |
infEdist_le_infEdist_cthickening_add :
infEdist x s ≤ infEdist x (cthickening δ s) + ENNReal.ofReal δ := by
refine le_of_forall_gt fun r h => ?_
simp_rw [← lt_tsub_iff_right, infEdist_lt_iff, mem_cthickening_iff] at h
obtain ⟨y, hy, hxy⟩ := h
exact infEdist_le_edist_add_infEdist.trans_lt
((ENNReal.add_lt_add_of_lt_of_le (hy.trans_lt ENNReal.ofReal_lt_top).ne hxy hy).trans_eq
(tsub_add_cancel_of_le <| le_self_add.trans (lt_tsub_iff_left.1 hxy).le)) | theorem | Topology | [
"Mathlib.Topology.MetricSpace.HausdorffDistance"
] | Mathlib/Topology/MetricSpace/Thickening.lean | infEdist_le_infEdist_cthickening_add | For the equality, see `infEdist_cthickening`. |
infEdist_le_infEdist_thickening_add :
infEdist x s ≤ infEdist x (thickening δ s) + ENNReal.ofReal δ :=
infEdist_le_infEdist_cthickening_add.trans <|
add_le_add_right (infEdist_anti <| thickening_subset_cthickening _ _) _ | theorem | Topology | [
"Mathlib.Topology.MetricSpace.HausdorffDistance"
] | Mathlib/Topology/MetricSpace/Thickening.lean | infEdist_le_infEdist_thickening_add | For the equality, see `infEdist_thickening`. |
@[simp]
thickening_thickening_subset (ε δ : ℝ) (s : Set α) :
thickening ε (thickening δ s) ⊆ thickening (ε + δ) s := by
obtain hε | hε := le_total ε 0
· simp only [thickening_of_nonpos hε, empty_subset]
obtain hδ | hδ := le_total δ 0
· simp only [thickening_of_nonpos hδ, thickening_empty, empty_subset]
intro x
simp_rw [mem_thickening_iff_exists_edist_lt, ENNReal.ofReal_add hε hδ]
exact fun ⟨y, ⟨z, hz, hy⟩, hx⟩ =>
⟨z, hz, (edist_triangle _ _ _).trans_lt <| ENNReal.add_lt_add hx hy⟩ | theorem | Topology | [
"Mathlib.Topology.MetricSpace.HausdorffDistance"
] | Mathlib/Topology/MetricSpace/Thickening.lean | thickening_thickening_subset | For the equality, see `thickening_thickening`. |
@[simp]
thickening_cthickening_subset (ε : ℝ) (hδ : 0 ≤ δ) (s : Set α) :
thickening ε (cthickening δ s) ⊆ thickening (ε + δ) s := by
obtain hε | hε := le_total ε 0
· simp only [thickening_of_nonpos hε, empty_subset]
intro x
simp_rw [mem_thickening_iff_exists_edist_lt, mem_cthickening_iff, ← infEdist_lt_iff,
ENNReal.ofReal_add hε hδ]
rintro ⟨y, hy, hxy⟩
exact infEdist_le_edist_add_infEdist.trans_lt
(ENNReal.add_lt_add_of_lt_of_le (hy.trans_lt ENNReal.ofReal_lt_top).ne hxy hy) | theorem | Topology | [
"Mathlib.Topology.MetricSpace.HausdorffDistance"
] | Mathlib/Topology/MetricSpace/Thickening.lean | thickening_cthickening_subset | For the equality, see `thickening_cthickening`. |
@[simp]
cthickening_thickening_subset (hε : 0 ≤ ε) (δ : ℝ) (s : Set α) :
cthickening ε (thickening δ s) ⊆ cthickening (ε + δ) s := by
obtain hδ | hδ := le_total δ 0
· simp only [thickening_of_nonpos hδ, cthickening_empty, empty_subset]
intro x
simp_rw [mem_cthickening_iff, ENNReal.ofReal_add hε hδ]
exact fun hx => infEdist_le_infEdist_thickening_add.trans (add_le_add_right hx _) | theorem | Topology | [
"Mathlib.Topology.MetricSpace.HausdorffDistance"
] | Mathlib/Topology/MetricSpace/Thickening.lean | cthickening_thickening_subset | For the equality, see `cthickening_thickening`. |
@[simp]
cthickening_cthickening_subset (hε : 0 ≤ ε) (hδ : 0 ≤ δ) (s : Set α) :
cthickening ε (cthickening δ s) ⊆ cthickening (ε + δ) s := by
intro x
simp_rw [mem_cthickening_iff, ENNReal.ofReal_add hε hδ]
exact fun hx => infEdist_le_infEdist_cthickening_add.trans (add_le_add_right hx _)
open scoped Function in -- required for scoped `on` notation | theorem | Topology | [
"Mathlib.Topology.MetricSpace.HausdorffDistance"
] | Mathlib/Topology/MetricSpace/Thickening.lean | cthickening_cthickening_subset | For the equality, see `cthickening_cthickening`. |
frontier_cthickening_disjoint (A : Set α) :
Pairwise (Disjoint on fun r : ℝ≥0 => frontier (cthickening r A)) := fun r₁ r₂ hr =>
((disjoint_singleton.2 <| by simpa).preimage _).mono (frontier_cthickening_subset _)
(frontier_cthickening_subset _) | theorem | Topology | [
"Mathlib.Topology.MetricSpace.HausdorffDistance"
] | Mathlib/Topology/MetricSpace/Thickening.lean | frontier_cthickening_disjoint | null |
thickening_ball [PseudoMetricSpace α] (x : α) (ε δ : ℝ) :
thickening ε (ball x δ) ⊆ ball x (ε + δ) := by
rw [← thickening_singleton, ← thickening_singleton]
apply thickening_thickening_subset | theorem | Topology | [
"Mathlib.Topology.MetricSpace.HausdorffDistance"
] | Mathlib/Topology/MetricSpace/Thickening.lean | thickening_ball | null |
IsClopen.of_thickening_subset_self {δ : ℝ} (hδ : 0 < δ) (hs : thickening δ s ⊆ s) :
IsClopen s := by
replace hs : thickening δ s = s := le_antisymm hs (self_subset_thickening hδ s)
refine ⟨?_, hs ▸ isOpen_thickening⟩
rw [← closure_subset_iff_isClosed, closure_eq_iInter_thickening]
exact Set.biInter_subset_of_mem hδ |>.trans_eq hs | lemma | Topology | [
"Mathlib.Topology.MetricSpace.HausdorffDistance"
] | Mathlib/Topology/MetricSpace/Thickening.lean | IsClopen.of_thickening_subset_self | null |
IsClopen.of_cthickening_subset_self {δ : ℝ} (hδ : 0 < δ) (hs : cthickening δ s ⊆ s) :
IsClopen s :=
.of_thickening_subset_self hδ <| (thickening_subset_cthickening δ s).trans hs | lemma | Topology | [
"Mathlib.Topology.MetricSpace.HausdorffDistance"
] | Mathlib/Topology/MetricSpace/Thickening.lean | IsClopen.of_cthickening_subset_self | null |
IsCompact.exists_thickening_image_subset
[PseudoEMetricSpace α] {β : Type*} [PseudoEMetricSpace β]
{f : α → β} {K : Set α} {U : Set β} (hK : IsCompact K) (ho : IsOpen U)
(hf : ∀ x ∈ K, ContinuousAt f x) (hKU : MapsTo f K U) :
∃ ε > 0, ∃ V ∈ 𝓝ˢ K, thickening ε (f '' V) ⊆ U := by
apply hK.induction_on (p := fun K ↦ ∃ ε > 0, ∃ V ∈ 𝓝ˢ K, thickening ε (f '' V) ⊆ U)
· use 1, by positivity, ∅, by simp, by simp
· exact fun s t hst ⟨ε, hε, V, hV, hthickening⟩ ↦ ⟨ε, hε, V, nhdsSet_mono hst hV, hthickening⟩
· rintro s t ⟨ε₁, hε₁, V₁, hV₁, hV₁thickening⟩ ⟨ε₂, hε₂, V₂, hV₂, hV₂thickening⟩
refine ⟨min ε₁ ε₂, by positivity, V₁ ∪ V₂, union_mem_nhdsSet hV₁ hV₂, ?_⟩
rw [image_union, thickening_union]
calc thickening (ε₁ ⊓ ε₂) (f '' V₁) ∪ thickening (ε₁ ⊓ ε₂) (f '' V₂)
_ ⊆ thickening ε₁ (f '' V₁) ∪ thickening ε₂ (f '' V₂) := by gcongr <;> norm_num
_ ⊆ U ∪ U := by gcongr
_ = U := union_self _
· intro x hx
have : {f x} ⊆ U := by rw [singleton_subset_iff]; exact hKU hx
obtain ⟨δ, hδ, hthick⟩ := (isCompact_singleton (x := f x)).exists_thickening_subset_open ho this
let V := f ⁻¹' (thickening (δ / 2) {f x})
have : V ∈ 𝓝 x := by
apply hf x hx
apply isOpen_thickening.mem_nhds
exact (self_subset_thickening (by positivity) _) rfl
refine ⟨K ∩ (interior V), inter_mem_nhdsWithin K (interior_mem_nhds.mpr this),
δ / 2, by positivity, V, by rw [← subset_interior_iff_mem_nhdsSet]; simp, ?_⟩
calc thickening (δ / 2) (f '' V)
_ ⊆ thickening (δ / 2) (thickening (δ / 2) {f x}) :=
thickening_subset_of_subset _ (image_preimage_subset f _)
_ ⊆ thickening ((δ / 2) + (δ / 2)) ({f x}) :=
thickening_thickening_subset (δ / 2) (δ / 2) {f x}
_ ⊆ U := by simp [hthick] | theorem | Topology | [
"Mathlib.Topology.MetricSpace.HausdorffDistance"
] | Mathlib/Topology/MetricSpace/Thickening.lean | IsCompact.exists_thickening_image_subset | null |
edist_def (f g : α →ᵤ β) :
edist f g = ⨆ x, edist (toFun f x) (toFun g x) :=
rfl | lemma | Topology | [
"Mathlib.Order.CompleteLattice.Group",
"Mathlib.Topology.ContinuousMap.Bounded.Basic",
"Mathlib.Topology.ContinuousMap.Compact",
"Mathlib.Topology.MetricSpace.Lipschitz",
"Mathlib.Topology.UniformSpace.UniformConvergenceTopology"
] | Mathlib/Topology/MetricSpace/UniformConvergence.lean | edist_def | null |
edist_le {f g : α →ᵤ β} {C : ℝ≥0∞} :
edist f g ≤ C ↔ ∀ x, edist (toFun f x) (toFun g x) ≤ C :=
iSup_le_iff | lemma | Topology | [
"Mathlib.Order.CompleteLattice.Group",
"Mathlib.Topology.ContinuousMap.Bounded.Basic",
"Mathlib.Topology.ContinuousMap.Compact",
"Mathlib.Topology.MetricSpace.Lipschitz",
"Mathlib.Topology.UniformSpace.UniformConvergenceTopology"
] | Mathlib/Topology/MetricSpace/UniformConvergence.lean | edist_le | null |
_root_.LipschitzWith.uniformEquicontinuous (f : α → γ → β) (K : ℝ≥0)
(h : ∀ c, LipschitzWith K (f c)) : UniformEquicontinuous f := by
rw [uniformEquicontinuous_iff_uniformContinuous]
rw [← lipschitzWith_ofFun_iff] at h
exact h.uniformContinuous | lemma | Topology | [
"Mathlib.Order.CompleteLattice.Group",
"Mathlib.Topology.ContinuousMap.Bounded.Basic",
"Mathlib.Topology.ContinuousMap.Compact",
"Mathlib.Topology.MetricSpace.Lipschitz",
"Mathlib.Topology.UniformSpace.UniformConvergenceTopology"
] | Mathlib/Topology/MetricSpace/UniformConvergence.lean | _root_.LipschitzWith.uniformEquicontinuous | The natural `EMetric` structure on `α →ᵤ β` given by `edist f g = ⨆ x, edist (f x) (g x)`. -/
noncomputable instance : PseudoEMetricSpace (α →ᵤ β) where
edist_self := by simp [edist_def]
edist_comm := by simp [edist_def, edist_comm]
edist_triangle f₁ f₂ f₃ := calc
⨆ x, edist (f₁ x) (f₃ x) ≤ ⨆ x, edist (f₁ x) (f₂ x) + edist (f₂ x) (f₃ x) :=
iSup_mono fun _ ↦ edist_triangle _ _ _
_ ≤ (⨆ x, edist (f₁ x) (f₂ x)) + (⨆ x, edist (f₂ x) (f₃ x)) := iSup_add_le _ _
toUniformSpace := inferInstance
uniformity_edist := by
suffices 𝓤 (α →ᵤ β) = comap (fun x ↦ edist x.1 x.2) (𝓝 0) by
simp [this, ENNReal.nhds_zero_basis.comap _ |>.eq_biInf, Set.Iio]
rw [ENNReal.nhds_zero_basis_Iic.comap _ |>.eq_biInf]
rw [UniformFun.hasBasis_uniformity_of_basis α β uniformity_basis_edist_le |>.eq_biInf]
simp [UniformFun.gen, edist_le, Set.Iic]
noncomputable instance {β : Type*} [EMetricSpace β] : EMetricSpace (α →ᵤ β) :=
.ofT0PseudoEMetricSpace _
lemma lipschitzWith_iff {f : γ → α →ᵤ β} {K : ℝ≥0} :
LipschitzWith K f ↔ ∀ c, LipschitzWith K (fun x ↦ toFun (f x) c) := by
simp [LipschitzWith, edist_le, forall_comm (α := α)]
lemma lipschitzWith_ofFun_iff {f : γ → α → β} {K : ℝ≥0} :
LipschitzWith K (fun x ↦ ofFun (f x)) ↔ ∀ c, LipschitzWith K (f · c) :=
lipschitzWith_iff
/-- If `f : α → γ → β` is a family of a functions, all of which are Lipschitz with the
same constant, then the family is uniformly equicontinuous. |
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