statement stringlengths 1 2.93k | proof stringlengths 0 19.2k | type stringclasses 13
values | symbolic_name stringlengths 1 131 | library stringlengths 4 62 | filename stringlengths 20 95 | imports listlengths 0 10 | deps listlengths 0 64 | docstring stringlengths 0 4.95k | source_url stringclasses 1
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thickenedIndicatorAux_lt_top {δ : ℝ} {E : Set α} {x : α} :
thickenedIndicatorAux δ E x < ∞ | lt_of_le_of_lt (thickenedIndicatorAux_le_one _ _ _) one_lt_top | theorem | thickenedIndicatorAux_lt_top | Topology.MetricSpace | Mathlib/Topology/MetricSpace/ThickenedIndicator.lean | [] | [
"Set",
"lt_of_le_of_lt",
"thickenedIndicatorAux",
"thickenedIndicatorAux_le_one"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
thickenedIndicatorAux_closure_eq (δ : ℝ) (E : Set α) :
thickenedIndicatorAux δ (closure E) = thickenedIndicatorAux δ E | by
simp +unfoldPartialApp only [thickenedIndicatorAux, infEDist_closure] | theorem | thickenedIndicatorAux_closure_eq | Topology.MetricSpace | Mathlib/Topology/MetricSpace/ThickenedIndicator.lean | [] | [
"Set",
"closure",
"thickenedIndicatorAux"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
thickenedIndicatorAux_one (δ : ℝ) (E : Set α) {x : α} (x_in_E : x ∈ E) :
thickenedIndicatorAux δ E x = 1 | by
simp [thickenedIndicatorAux, infEDist_zero_of_mem x_in_E, tsub_zero] | theorem | thickenedIndicatorAux_one | Topology.MetricSpace | Mathlib/Topology/MetricSpace/ThickenedIndicator.lean | [] | [
"Set",
"thickenedIndicatorAux",
"tsub_zero"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
thickenedIndicatorAux_one_of_mem_closure (δ : ℝ) (E : Set α) {x : α}
(x_mem : x ∈ closure E) : thickenedIndicatorAux δ E x = 1 | by
rw [← thickenedIndicatorAux_closure_eq, thickenedIndicatorAux_one δ (closure E) x_mem] | theorem | thickenedIndicatorAux_one_of_mem_closure | Topology.MetricSpace | Mathlib/Topology/MetricSpace/ThickenedIndicator.lean | [] | [
"Set",
"closure",
"thickenedIndicatorAux",
"thickenedIndicatorAux_closure_eq",
"thickenedIndicatorAux_one"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
thickenedIndicatorAux_zero {δ : ℝ} (δ_pos : 0 < δ) (E : Set α) {x : α}
(x_out : x ∉ thickening δ E) : thickenedIndicatorAux δ E x = 0 | by
rw [thickening, mem_setOf_eq, not_lt] at x_out
unfold thickenedIndicatorAux
apply le_antisymm _ bot_le
have key := tsub_le_tsub
(@rfl _ (1 : ℝ≥0∞)).le (ENNReal.div_le_div x_out (@rfl _ (ENNReal.ofReal δ : ℝ≥0∞)).le)
rw [ENNReal.div_self (ne_of_gt (ENNReal.ofReal_pos.mpr δ_pos)) ofReal_ne_top] at key
... | theorem | thickenedIndicatorAux_zero | Topology.MetricSpace | Mathlib/Topology/MetricSpace/ThickenedIndicator.lean | [] | [
"ENNReal.div_le_div",
"ENNReal.div_self",
"ENNReal.ofReal",
"Set",
"le_antisymm",
"not_lt",
"thickenedIndicatorAux",
"tsub_le_tsub",
"tsub_self"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
thickenedIndicatorAux_mono {δ₁ δ₂ : ℝ} (hle : δ₁ ≤ δ₂) (E : Set α) :
thickenedIndicatorAux δ₁ E ≤ thickenedIndicatorAux δ₂ E | fun _ => tsub_le_tsub (@rfl ℝ≥0∞ 1).le (ENNReal.div_le_div rfl.le (ofReal_le_ofReal hle)) | theorem | thickenedIndicatorAux_mono | Topology.MetricSpace | Mathlib/Topology/MetricSpace/ThickenedIndicator.lean | [] | [
"ENNReal.div_le_div",
"Set",
"thickenedIndicatorAux",
"tsub_le_tsub"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
indicator_le_thickenedIndicatorAux (δ : ℝ) (E : Set α) :
(E.indicator fun _ => (1 : ℝ≥0∞)) ≤ thickenedIndicatorAux δ E | by
intro a
by_cases h : a ∈ E
· simp only [h, indicator_of_mem, thickenedIndicatorAux_one δ E h, le_refl]
· simp only [h, indicator_of_notMem, not_false_iff, zero_le] | theorem | indicator_le_thickenedIndicatorAux | Topology.MetricSpace | Mathlib/Topology/MetricSpace/ThickenedIndicator.lean | [] | [
"Set",
"by_cases",
"le_refl",
"thickenedIndicatorAux",
"thickenedIndicatorAux_one"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
thickenedIndicatorAux_subset (δ : ℝ) {E₁ E₂ : Set α} (subset : E₁ ⊆ E₂) :
thickenedIndicatorAux δ E₁ ≤ thickenedIndicatorAux δ E₂ | fun _ => tsub_le_tsub (@rfl ℝ≥0∞ 1).le (ENNReal.div_le_div (infEDist_anti subset) rfl.le) | theorem | thickenedIndicatorAux_subset | Topology.MetricSpace | Mathlib/Topology/MetricSpace/ThickenedIndicator.lean | [] | [
"ENNReal.div_le_div",
"Set",
"thickenedIndicatorAux",
"tsub_le_tsub"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
thickenedIndicatorAux_mono_infEDist (δ : ℝ) {E : Set α} {x y : α}
(h : infEDist x E ≤ infEDist y E) :
thickenedIndicatorAux δ E y ≤ thickenedIndicatorAux δ E x | by
simp only [thickenedIndicatorAux]
rcases le_total (infEDist x E / ENNReal.ofReal δ) 1 with hle | hle
· rw [ENNReal.sub_le_sub_iff_left hle (by simp)]
gcongr
· rw [tsub_eq_zero_of_le hle, tsub_eq_zero_of_le]
exact hle.trans (by gcongr)
@[deprecated (since := "2026-01-08")]
alias thickenedIndicatorAux... | lemma | thickenedIndicatorAux_mono_infEDist | Topology.MetricSpace | Mathlib/Topology/MetricSpace/ThickenedIndicator.lean | [] | [
"ENNReal.ofReal",
"ENNReal.sub_le_sub_iff_left",
"Set",
"le_total",
"thickenedIndicatorAux"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
thickenedIndicatorAux_tendsto_indicator_closure {δseq : ℕ → ℝ}
(δseq_lim : Tendsto δseq atTop (𝓝 0)) (E : Set α) :
Tendsto (fun n => thickenedIndicatorAux (δseq n) E) atTop
(𝓝 (indicator (closure E) fun _ => (1 : ℝ≥0∞))) | by
rw [tendsto_pi_nhds]
intro x
by_cases x_mem_closure : x ∈ closure E
· simp_rw [thickenedIndicatorAux_one_of_mem_closure _ E x_mem_closure]
rw [show (indicator (closure E) fun _ => (1 : ℝ≥0∞)) x = 1 by
simp only [x_mem_closure, indicator_of_mem]]
exact tendsto_const_nhds
· rw [show (closure ... | theorem | thickenedIndicatorAux_tendsto_indicator_closure | Topology.MetricSpace | Mathlib/Topology/MetricSpace/ThickenedIndicator.lean | [] | [
"Metric.tendsto_nhds",
"Real.norm_eq_abs",
"Set",
"by_cases",
"closure",
"le_antisymm",
"not_lt",
"tendsto_atTop_of_eventually_const",
"tendsto_const_nhds",
"tendsto_pi_nhds",
"thickenedIndicatorAux",
"thickenedIndicatorAux_mono",
"thickenedIndicatorAux_one_of_mem_closure",
"thickenedIndic... | As the thickening radius δ tends to 0, the δ-thickened indicator of a set E (in α) tends
pointwise (i.e., w.r.t. the product topology on `α → ℝ≥0∞`) to the indicator function of the
closure of E.
This statement is for the unbundled `ℝ≥0∞`-valued functions `thickenedIndicatorAux δ E`, see
`thickenedIndicator_tendsto_in... | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
thickenedIndicator {δ : ℝ} (δ_pos : 0 < δ) (E : Set α) : α →ᵇ ℝ≥0 | where
toFun := fun x : α => (thickenedIndicatorAux δ E x).toNNReal
continuous_toFun := by
apply ContinuousOn.comp_continuous continuousOn_toNNReal
(continuous_thickenedIndicatorAux δ_pos E)
intro x
exact (lt_of_le_of_lt (@thickenedIndicatorAux_le_one _ _ δ E x) one_lt_top).ne
map_bounded' := by
... | def | thickenedIndicator | Topology.MetricSpace | Mathlib/Topology/MetricSpace/ThickenedIndicator.lean | [] | [
"ContinuousOn.comp_continuous",
"NNReal.abs_eq",
"NNReal.dist_eq",
"Set",
"continuous_thickenedIndicatorAux",
"lt_of_le_of_lt",
"one_add_one_eq_two",
"thickenedIndicatorAux",
"thickenedIndicatorAux_le_one",
"trans"
] | The `δ`-thickened indicator of a set `E` is the function that equals `1` on `E`
and `0` outside a `δ`-thickening of `E` and interpolates (continuously) between
these values using `infEDist _ E`.
`thickenedIndicator` is the (bundled) bounded continuous function with `ℝ≥0`-values.
See `thickenedIndicatorAux` for the unb... | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
thickenedIndicator.coeFn_eq_comp {δ : ℝ} (δ_pos : 0 < δ) (E : Set α) :
⇑(thickenedIndicator δ_pos E) = ENNReal.toNNReal ∘ thickenedIndicatorAux δ E | rfl | theorem | thickenedIndicator.coeFn_eq_comp | Topology.MetricSpace | Mathlib/Topology/MetricSpace/ThickenedIndicator.lean | [] | [
"ENNReal.toNNReal",
"Set",
"thickenedIndicator",
"thickenedIndicatorAux"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
thickenedIndicator_le_one {δ : ℝ} (δ_pos : 0 < δ) (E : Set α) (x : α) :
thickenedIndicator δ_pos E x ≤ 1 | by
rw [thickenedIndicator.coeFn_eq_comp]
simpa using (toNNReal_le_toNNReal (by finiteness) one_ne_top).mpr
(thickenedIndicatorAux_le_one δ E x) | theorem | thickenedIndicator_le_one | Topology.MetricSpace | Mathlib/Topology/MetricSpace/ThickenedIndicator.lean | [] | [
"Set",
"thickenedIndicator",
"thickenedIndicator.coeFn_eq_comp",
"thickenedIndicatorAux_le_one"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
thickenedIndicator_one_of_mem_closure {δ : ℝ} (δ_pos : 0 < δ) (E : Set α) {x : α}
(x_mem : x ∈ closure E) : thickenedIndicator δ_pos E x = 1 | by
rw [thickenedIndicator_apply, thickenedIndicatorAux_one_of_mem_closure δ E x_mem, toNNReal_one] | theorem | thickenedIndicator_one_of_mem_closure | Topology.MetricSpace | Mathlib/Topology/MetricSpace/ThickenedIndicator.lean | [] | [
"Set",
"closure",
"thickenedIndicator",
"thickenedIndicatorAux_one_of_mem_closure"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
one_le_thickenedIndicator_apply' {X : Type _} [PseudoEMetricSpace X]
{δ : ℝ} (δ_pos : 0 < δ) {F : Set X} {x : X} (hxF : x ∈ closure F) :
1 ≤ thickenedIndicator δ_pos F x | by
rw [thickenedIndicator_one_of_mem_closure δ_pos F hxF] | lemma | one_le_thickenedIndicator_apply' | Topology.MetricSpace | Mathlib/Topology/MetricSpace/ThickenedIndicator.lean | [] | [
"PseudoEMetricSpace",
"Set",
"closure",
"thickenedIndicator",
"thickenedIndicator_one_of_mem_closure"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
one_le_thickenedIndicator_apply (X : Type _) [PseudoEMetricSpace X]
{δ : ℝ} (δ_pos : 0 < δ) {F : Set X} {x : X} (hxF : x ∈ F) :
1 ≤ thickenedIndicator δ_pos F x | one_le_thickenedIndicator_apply' δ_pos (subset_closure hxF) | lemma | one_le_thickenedIndicator_apply | Topology.MetricSpace | Mathlib/Topology/MetricSpace/ThickenedIndicator.lean | [] | [
"PseudoEMetricSpace",
"Set",
"one_le_thickenedIndicator_apply'",
"subset_closure",
"thickenedIndicator"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
thickenedIndicator_one {δ : ℝ} (δ_pos : 0 < δ) (E : Set α) {x : α} (x_in_E : x ∈ E) :
thickenedIndicator δ_pos E x = 1 | thickenedIndicator_one_of_mem_closure _ _ (subset_closure x_in_E) | theorem | thickenedIndicator_one | Topology.MetricSpace | Mathlib/Topology/MetricSpace/ThickenedIndicator.lean | [] | [
"Set",
"subset_closure",
"thickenedIndicator",
"thickenedIndicator_one_of_mem_closure"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
thickenedIndicator_zero {δ : ℝ} (δ_pos : 0 < δ) (E : Set α) {x : α}
(x_out : x ∉ thickening δ E) : thickenedIndicator δ_pos E x = 0 | by
rw [thickenedIndicator_apply, thickenedIndicatorAux_zero δ_pos E x_out, toNNReal_zero] | theorem | thickenedIndicator_zero | Topology.MetricSpace | Mathlib/Topology/MetricSpace/ThickenedIndicator.lean | [] | [
"Set",
"thickenedIndicator",
"thickenedIndicatorAux_zero"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
indicator_le_thickenedIndicator {δ : ℝ} (δ_pos : 0 < δ) (E : Set α) :
(E.indicator fun _ => (1 : ℝ≥0)) ≤ thickenedIndicator δ_pos E | by
intro a
by_cases h : a ∈ E
· simp only [h, indicator_of_mem, thickenedIndicator_one δ_pos E h, le_refl]
· simp only [h, indicator_of_notMem, not_false_iff, zero_le] | theorem | indicator_le_thickenedIndicator | Topology.MetricSpace | Mathlib/Topology/MetricSpace/ThickenedIndicator.lean | [] | [
"Set",
"by_cases",
"le_refl",
"thickenedIndicator",
"thickenedIndicator_one"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
thickenedIndicator_mono {δ₁ δ₂ : ℝ} (δ₁_pos : 0 < δ₁) (δ₂_pos : 0 < δ₂) (hle : δ₁ ≤ δ₂)
(E : Set α) : ⇑(thickenedIndicator δ₁_pos E) ≤ thickenedIndicator δ₂_pos E | by
intro x
apply (toNNReal_le_toNNReal (by finiteness) (by finiteness)).mpr
apply thickenedIndicatorAux_mono hle | theorem | thickenedIndicator_mono | Topology.MetricSpace | Mathlib/Topology/MetricSpace/ThickenedIndicator.lean | [] | [
"Set",
"thickenedIndicator",
"thickenedIndicatorAux_mono"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
thickenedIndicator_subset {δ : ℝ} (δ_pos : 0 < δ) {E₁ E₂ : Set α} (subset : E₁ ⊆ E₂) :
⇑(thickenedIndicator δ_pos E₁) ≤ thickenedIndicator δ_pos E₂ | fun x =>
(toNNReal_le_toNNReal (by finiteness) (by finiteness)).mpr
(thickenedIndicatorAux_subset δ subset x) | theorem | thickenedIndicator_subset | Topology.MetricSpace | Mathlib/Topology/MetricSpace/ThickenedIndicator.lean | [] | [
"Set",
"thickenedIndicator",
"thickenedIndicatorAux_subset"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
thickenedIndicator_mono_infEDist {δ : ℝ} (δ_pos : 0 < δ) {E : Set α} {x y : α}
(h : infEDist x E ≤ infEDist y E) :
thickenedIndicator δ_pos E y ≤ thickenedIndicator δ_pos E x | by
simp only [thickenedIndicator_apply]
gcongr
· finiteness
· exact thickenedIndicatorAux_mono_infEDist δ h
@[deprecated (since := "2026-01-08")]
alias thickenedIndicator_mono_infEdist := thickenedIndicator_mono_infEDist | lemma | thickenedIndicator_mono_infEDist | Topology.MetricSpace | Mathlib/Topology/MetricSpace/ThickenedIndicator.lean | [] | [
"Set",
"thickenedIndicator",
"thickenedIndicatorAux_mono_infEDist"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
thickenedIndicator_tendsto_indicator_closure {δseq : ℕ → ℝ} (δseq_pos : ∀ n, 0 < δseq n)
(δseq_lim : Tendsto δseq atTop (𝓝 0)) (E : Set α) :
Tendsto (fun n : ℕ => ((↑) : (α →ᵇ ℝ≥0) → α → ℝ≥0) (thickenedIndicator (δseq_pos n) E)) atTop
(𝓝 (indicator (closure E) fun _ => (1 : ℝ≥0))) | by
have key := thickenedIndicatorAux_tendsto_indicator_closure δseq_lim E
rw [tendsto_pi_nhds] at *
intro x
rw [show indicator (closure E) (fun _ => (1 : ℝ≥0)) x =
(indicator (closure E) (fun _ => (1 : ℝ≥0∞)) x).toNNReal
by refine (congr_fun (comp_indicator_const 1 ENNReal.toNNReal toNNReal_zero) ... | theorem | thickenedIndicator_tendsto_indicator_closure | Topology.MetricSpace | Mathlib/Topology/MetricSpace/ThickenedIndicator.lean | [] | [
"ENNReal.toNNReal",
"Set",
"by_cases",
"closure",
"symm",
"tendsto_pi_nhds",
"thickenedIndicator",
"thickenedIndicatorAux_tendsto_indicator_closure"
] | As the thickening radius δ tends to 0, the δ-thickened indicator of a set E (in α) tends
pointwise to the indicator function of the closure of E.
Note: This version is for the bundled bounded continuous functions, but the topology is not
the topology on `α →ᵇ ℝ≥0`. Coercions to functions `α → ℝ≥0` are done first, so t... | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
lipschitzWith_thickenedIndicator {δ : ℝ} (δ_pos : 0 < δ) (E : Set α) :
LipschitzWith δ.toNNReal⁻¹ (thickenedIndicator δ_pos E) | by
intro x y
wlog h : infEDist x E ≤ infEDist y E generalizing x y
· specialize this y x (le_of_not_ge h)
rwa [edist_comm, edist_comm x]
simp_rw [edist_dist, NNReal.dist_eq, thickenedIndicator_apply, coe_toNNReal_eq_toReal]
rw [← ENNReal.toReal_sub_of_le (thickenedIndicatorAux_mono_infEDist _ h) (by finit... | lemma | lipschitzWith_thickenedIndicator | Topology.MetricSpace | Mathlib/Topology/MetricSpace/ThickenedIndicator.lean | [] | [
"ENNReal.coe_inv",
"ENNReal.ofReal",
"ENNReal.sub_mul",
"ENNReal.sub_sub_sub_cancel_left",
"ENNReal.toReal_sub_of_le",
"LipschitzWith",
"NNReal.dist_eq",
"Set",
"by_cases",
"div_eq_mul_inv",
"edist_dist",
"le_of_not_ge",
"mul_comm",
"thickenedIndicator",
"thickenedIndicatorAux",
"thick... | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
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 ... | lemma | mulIndicator_thickening_eventually_eq_mulIndicator_closure | Topology.MetricSpace | Mathlib/Topology/MetricSpace/ThickenedIndicator.lean | [] | [
"Metric.thickening",
"Set",
"by_cases",
"closure",
"mem_nhdsWithin_of_mem_nhds",
"self_mem_nhdsWithin"
] | 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. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
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] w... | lemma | mulIndicator_cthickening_eventually_eq_mulIndicator_closure | Topology.MetricSpace | Mathlib/Topology/MetricSpace/ThickenedIndicator.lean | [] | [
"Metric.cthickening",
"Set",
"by_cases",
"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. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
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 | tendsto_mulIndicator_thickening_mulIndicator_closure | Topology.MetricSpace | Mathlib/Topology/MetricSpace/ThickenedIndicator.lean | [] | [
"Metric.thickening",
"Set",
"closure",
"mulIndicator_thickening_eventually_eq_mulIndicator_closure",
"tendsto_const_nhds",
"tendsto_pi_nhds"
] | The multiplicative indicators of δ-thickenings of a set tend pointwise to the multiplicative
indicator of the set, as δ>0 tends to zero. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
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 | tendsto_mulIndicator_cthickening_mulIndicator_closure | Topology.MetricSpace | Mathlib/Topology/MetricSpace/ThickenedIndicator.lean | [] | [
"Metric.cthickening",
"Set",
"closure",
"mulIndicator_cthickening_eventually_eq_mulIndicator_closure",
"tendsto_const_nhds",
"tendsto_pi_nhds"
] | The multiplicative indicators of closed δ-thickenings of a set tend pointwise to the
multiplicative indicator of the set, as δ tends to zero. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
thickening (δ : ℝ) (E : Set α) : Set α | { x : α | infEDist x E < ENNReal.ofReal δ } | def | Metric.thickening | Topology.MetricSpace | Mathlib/Topology/MetricSpace/Thickening.lean | [] | [
"ENNReal.ofReal",
"Set"
] | 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`. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
mem_thickening_iff_infEDist_lt : x ∈ thickening δ s ↔ infEDist x s < ENNReal.ofReal δ | Iff.rfl
@[deprecated (since := "2026-01-08")]
alias mem_thickening_iff_infEdist_lt := mem_thickening_iff_infEDist_lt | theorem | Metric.mem_thickening_iff_infEDist_lt | Topology.MetricSpace | Mathlib/Topology/MetricSpace/Thickening.lean | [] | [
"ENNReal.ofReal"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
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 := "2026-01-08")]
alias eventually_notMem_thickening_of_infEd... | lemma | Metric.eventually_notMem_thickening_of_infEDist_pos | Topology.MetricSpace | Mathlib/Topology/MetricSpace/Thickening.lean | [] | [
"ENNReal.ofReal_le_ofReal",
"Metric.thickening",
"Set",
"closure",
"not_lt",
"trans"
] | 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`. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
thickening_eq_preimage_infEDist (δ : ℝ) (E : Set α) :
thickening δ E = (infEDist · E) ⁻¹' Iio (ENNReal.ofReal δ) | rfl
@[deprecated (since := "2026-01-08")]
alias thickening_eq_preimage_infEdist := thickening_eq_preimage_infEDist | theorem | Metric.thickening_eq_preimage_infEDist | Topology.MetricSpace | Mathlib/Topology/MetricSpace/Thickening.lean | [] | [
"ENNReal.ofReal",
"Set"
] | The (open) thickening equals the preimage of an open interval under `Metric.infEDist`. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
isOpen_thickening {δ : ℝ} {E : Set α} : IsOpen (thickening δ E) | Continuous.isOpen_preimage continuous_infEDist _ isOpen_Iio | theorem | Metric.isOpen_thickening | Topology.MetricSpace | Mathlib/Topology/MetricSpace/Thickening.lean | [] | [
"IsOpen",
"Set"
] | The (open) thickening is an open set. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
thickening_empty (δ : ℝ) : thickening δ (∅ : Set α) = ∅ | by
simp only [thickening, setOf_false, infEDist_empty, not_top_lt] | theorem | Metric.thickening_empty | Topology.MetricSpace | Mathlib/Topology/MetricSpace/Thickening.lean | [] | [
"Set",
"not_top_lt"
] | The (open) thickening of the empty set is empty. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
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 | Metric.thickening_of_nonpos | Topology.MetricSpace | Mathlib/Topology/MetricSpace/Thickening.lean | [] | [
"Set"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
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 | Metric.thickening_mono | Topology.MetricSpace | Mathlib/Topology/MetricSpace/Thickening.lean | [] | [
"ENNReal.ofReal_le_ofReal",
"Set"
] | The (open) thickening `Metric.thickening δ E` of a fixed subset `E` is an increasing function of
the thickening radius `δ`. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
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 | Metric.thickening_subset_of_subset | Topology.MetricSpace | Mathlib/Topology/MetricSpace/Thickening.lean | [] | [
"Set",
"lt_of_le_of_lt"
] | The (open) thickening `Metric.thickening δ E` with a fixed thickening radius `δ` is
an increasing function of the subset `E`. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
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 | Metric.mem_thickening_iff_exists_edist_lt | Topology.MetricSpace | Mathlib/Topology/MetricSpace/Thickening.lean | [] | [
"ENNReal.ofReal",
"Set"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
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 | theorem | Metric.frontier_thickening_subset | Topology.MetricSpace | Mathlib/Topology/MetricSpace/Thickening.lean | [] | [
"ENNReal.ofReal",
"Set",
"continuous_const",
"frontier",
"frontier_lt_subset_eq"
] | The frontier of the (open) thickening of a set is contained in an `Metric.infEDist` level
set. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
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_ofR... | theorem | Metric.frontier_thickening_disjoint | Topology.MetricSpace | Mathlib/Topology/MetricSpace/Thickening.lean | [] | [
"Disjoint",
"ENNReal.toReal",
"ENNReal.toReal_ofReal",
"Pairwise",
"Set",
"frontier",
"le_total",
"pairwise_disjoint_on"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
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 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 <| Metric.infEDist_le_edist_of_mem x_in_E | lemma | Metric.subset_compl_thickening_compl_thickening_self | Topology.MetricSpace | Mathlib/Topology/MetricSpace/Thickening.lean | [] | [
"Metric.infEDist_le_edist_of_mem",
"Set",
"le_trans",
"not_lt"
] | Any set is contained in the complement of the δ-thickening of the complement of its
δ-thickening. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
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 | lemma | Metric.thickening_compl_thickening_self_subset_compl | Topology.MetricSpace | Mathlib/Topology/MetricSpace/Thickening.lean | [] | [
"Set",
"compl_compl"
] | The δ-thickening of the complement of the δ-thickening of a set is contained in the complement
of the set. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
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 | Metric.mem_thickening_iff_infDist_lt | Topology.MetricSpace | Mathlib/Topology/MetricSpace/Thickening.lean | [] | [
"Set"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
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] | theorem | Metric.mem_thickening_iff | Topology.MetricSpace | Mathlib/Topology/MetricSpace/Thickening.lean | [] | [
"ENNReal.ofReal",
"Set",
"dist_edist",
"edist_ne_top"
] | 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`. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
thickening_singleton (δ : ℝ) (x : X) : thickening δ ({x} : Set X) = ball x δ | by
ext
simp [mem_thickening_iff] | theorem | Metric.thickening_singleton | Topology.MetricSpace | Mathlib/Topology/MetricSpace/Thickening.lean | [] | [
"Set"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
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 | Metric.ball_subset_thickening | Topology.MetricSpace | Mathlib/Topology/MetricSpace/Thickening.lean | [] | [
"Set"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
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 | Metric.thickening_eq_biUnion_ball | Topology.MetricSpace | Mathlib/Topology/MetricSpace/Thickening.lean | [] | [
"Set"
] | 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`. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
_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 := by grw [(mem_thickening_iff_infDist_lt ⟨x, hx⟩).1 hy] | theorem | Bornology.IsBounded.thickening | Topology.MetricSpace | Mathlib/Topology/MetricSpace/Thickening.lean | [] | [
"Set"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
cthickening (δ : ℝ) (E : Set α) : Set α | { x : α | infEDist x E ≤ ENNReal.ofReal δ } | def | Metric.cthickening | Topology.MetricSpace | Mathlib/Topology/MetricSpace/Thickening.lean | [] | [
"ENNReal.ofReal",
"Set"
] | 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`. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
mem_cthickening_iff : x ∈ cthickening δ s ↔ infEDist x s ≤ ENNReal.ofReal δ | Iff.rfl | theorem | Metric.mem_cthickening_iff | Topology.MetricSpace | Mathlib/Topology/MetricSpace/Thickening.lean | [] | [
"ENNReal.ofReal"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
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 := "2026-01-08")]
alias eventually_notMem_cthickening_of_i... | lemma | Metric.eventually_notMem_cthickening_of_infEDist_pos | Topology.MetricSpace | Mathlib/Topology/MetricSpace/Thickening.lean | [] | [
"Metric.cthickening",
"Set",
"closure",
"not_le",
"trans"
] | 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`. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
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 | Metric.mem_cthickening_of_edist_le | Topology.MetricSpace | Mathlib/Topology/MetricSpace/Thickening.lean | [] | [
"ENNReal.ofReal",
"Set",
"trans"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
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 | Metric.mem_cthickening_of_dist_le | Topology.MetricSpace | Mathlib/Topology/MetricSpace/Thickening.lean | [] | [
"ENNReal.ofReal_le_ofReal",
"PseudoMetricSpace",
"Set",
"edist_dist"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
cthickening_eq_preimage_infEDist (δ : ℝ) (E : Set α) :
cthickening δ E = (fun x => infEDist x E) ⁻¹' Iic (ENNReal.ofReal δ) | rfl
@[deprecated (since := "2026-01-08")]
alias cthickening_eq_preimage_infEdist := cthickening_eq_preimage_infEDist | theorem | Metric.cthickening_eq_preimage_infEDist | Topology.MetricSpace | Mathlib/Topology/MetricSpace/Thickening.lean | [] | [
"ENNReal.ofReal",
"Set"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
isClosed_cthickening {δ : ℝ} {E : Set α} : IsClosed (cthickening δ E) | IsClosed.preimage continuous_infEDist isClosed_Iic | theorem | Metric.isClosed_cthickening | Topology.MetricSpace | Mathlib/Topology/MetricSpace/Thickening.lean | [] | [
"IsClosed",
"IsClosed.preimage",
"Set",
"isClosed_Iic"
] | The closed thickening is a closed set. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
cthickening_empty (δ : ℝ) : cthickening δ (∅ : Set α) = ∅ | by
simp only [cthickening, ENNReal.ofReal_ne_top, setOf_false, infEDist_empty, top_le_iff] | theorem | Metric.cthickening_empty | Topology.MetricSpace | Mathlib/Topology/MetricSpace/Thickening.lean | [] | [
"ENNReal.ofReal_ne_top",
"Set",
"top_le_iff"
] | The closed thickening of the empty set is empty. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
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 | Metric.cthickening_of_nonpos | Topology.MetricSpace | Mathlib/Topology/MetricSpace/Thickening.lean | [] | [
"Set",
"closure"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
cthickening_zero (E : Set α) : cthickening 0 E = closure E | cthickening_of_nonpos le_rfl E | theorem | Metric.cthickening_zero | Topology.MetricSpace | Mathlib/Topology/MetricSpace/Thickening.lean | [] | [
"Set",
"closure",
"le_rfl"
] | The closed thickening with radius zero is the closure of the set. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
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 | Metric.cthickening_max_zero | Topology.MetricSpace | Mathlib/Topology/MetricSpace/Thickening.lean | [] | [
"Set",
"le_total"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
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)) | theorem | Metric.cthickening_mono | Topology.MetricSpace | Mathlib/Topology/MetricSpace/Thickening.lean | [] | [
"ENNReal.ofReal_le_ofReal",
"Set"
] | The closed thickening `Metric.cthickening δ E` of a fixed subset `E` is an increasing function
of the thickening radius `δ`. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
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 | Metric.cthickening_singleton | Topology.MetricSpace | Mathlib/Topology/MetricSpace/Thickening.lean | [] | [
"ENNReal.ofReal_le_ofReal_iff",
"PseudoMetricSpace",
"Set",
"edist_dist"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
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 | Metric.closedBall_subset_cthickening_singleton | Topology.MetricSpace | Mathlib/Topology/MetricSpace/Thickening.lean | [] | [
"PseudoMetricSpace",
"Set",
"lt_or_ge"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
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 | Metric.cthickening_subset_of_subset | Topology.MetricSpace | Mathlib/Topology/MetricSpace/Thickening.lean | [] | [
"Set",
"le_trans"
] | The closed thickening `Metric.cthickening δ E` with a fixed thickening radius `δ` is
an increasing function of the subset `E`. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
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 | Metric.cthickening_subset_thickening | Topology.MetricSpace | Mathlib/Topology/MetricSpace/Thickening.lean | [] | [
"ENNReal.ofReal_lt_ofReal_iff",
"Set",
"lt_of_le_of_lt"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
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 | Metric.cthickening_subset_thickening' | Topology.MetricSpace | Mathlib/Topology/MetricSpace/Thickening.lean | [] | [
"ENNReal.ofReal_lt_ofReal_iff",
"Set",
"lt_of_le_of_lt"
] | 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. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
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 | Metric.thickening_subset_cthickening | Topology.MetricSpace | Mathlib/Topology/MetricSpace/Thickening.lean | [] | [
"Set"
] | The open thickening `Metric.thickening δ E` is contained in the closed thickening
`Metric.cthickening δ E` with the same radius. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
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 | Metric.thickening_subset_cthickening_of_le | Topology.MetricSpace | Mathlib/Topology/MetricSpace/Thickening.lean | [] | [
"Set",
"trans"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
_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 | Bornology.IsBounded.cthickening | Topology.MetricSpace | Mathlib/Topology/MetricSpace/Thickening.lean | [] | [
"PseudoMetricSpace",
"Set",
"lt_add_one"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
_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 | IsCompact.cthickening | Topology.MetricSpace | Mathlib/Topology/MetricSpace/Thickening.lean | [] | [
"IsCompact",
"ProperSpace",
"PseudoMetricSpace",
"Set"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
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 | Metric.thickening_subset_interior_cthickening | Topology.MetricSpace | Mathlib/Topology/MetricSpace/Thickening.lean | [] | [
"Set",
"interior",
"interior_mono",
"trans"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
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 | Metric.closure_thickening_subset_cthickening | Topology.MetricSpace | Mathlib/Topology/MetricSpace/Thickening.lean | [] | [
"Set",
"closure",
"closure_mono",
"trans"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
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 | Metric.closure_subset_cthickening | Topology.MetricSpace | Mathlib/Topology/MetricSpace/Thickening.lean | [] | [
"Set",
"closure",
"min_le_left",
"min_le_right"
] | The closed thickening of a set contains the closure of the set. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
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 | Metric.closure_subset_thickening | Topology.MetricSpace | Mathlib/Topology/MetricSpace/Thickening.lean | [] | [
"Set",
"closure"
] | The (open) thickening of a set contains the closure of the set. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
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 | Metric.self_subset_thickening | Topology.MetricSpace | Mathlib/Topology/MetricSpace/Thickening.lean | [] | [
"Set",
"subset_closure",
"trans"
] | A set is contained in its own (open) thickening. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
self_subset_cthickening {δ : ℝ} (E : Set α) : E ⊆ cthickening δ E | subset_closure.trans (closure_subset_cthickening δ E) | theorem | Metric.self_subset_cthickening | Topology.MetricSpace | Mathlib/Topology/MetricSpace/Thickening.lean | [] | [
"Set"
] | A set is contained in its own closed thickening. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
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 | Metric.thickening_mem_nhdsSet | Topology.MetricSpace | Mathlib/Topology/MetricSpace/Thickening.lean | [] | [
"Set"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
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 _ _) | theorem | Metric.cthickening_mem_nhdsSet | Topology.MetricSpace | Mathlib/Topology/MetricSpace/Thickening.lean | [] | [
"Set"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
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] | theorem | Metric.thickening_union | Topology.MetricSpace | Mathlib/Topology/MetricSpace/Thickening.lean | [] | [
"Set"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
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] | theorem | Metric.cthickening_union | Topology.MetricSpace | Mathlib/Topology/MetricSpace/Thickening.lean | [] | [
"Set"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
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 | Metric.thickening_iUnion | Topology.MetricSpace | Mathlib/Topology/MetricSpace/Thickening.lean | [] | [
"Set"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
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 | Metric.thickening_biUnion | Topology.MetricSpace | Mathlib/Topology/MetricSpace/Thickening.lean | [] | [
"Set"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
ediam_cthickening_le (ε : ℝ≥0) :
ediam (cthickening ε s) ≤ ediam s + 2 * ε | by
refine ediam_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
... | theorem | Metric.ediam_cthickening_le | Topology.MetricSpace | Mathlib/Topology/MetricSpace/Thickening.lean | [] | [
"ENNReal.le_of_forall_pos_le_add",
"ENNReal.ofReal_coe_nnreal",
"edist_triangle_right",
"two_mul"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
ediam_thickening_le (ε : ℝ≥0) : ediam (thickening ε s) ≤ ediam s + 2 * ε | (ediam_mono <| thickening_subset_cthickening _ _).trans <| ediam_cthickening_le _ | theorem | Metric.ediam_thickening_le | Topology.MetricSpace | Mathlib/Topology/MetricSpace/Thickening.lean | [] | [
"trans"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
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 ▸ ediam_mono (self_subset_cthickening _)
· simp [mul_eq_top]
· simp [diam] | theorem | Metric.diam_cthickening_le | Topology.MetricSpace | Mathlib/Topology/MetricSpace/Thickening.lean | [] | [
"PseudoMetricSpace",
"Set",
"top_unique"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
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)]
... | theorem | Metric.diam_thickening_le | Topology.MetricSpace | Mathlib/Topology/MetricSpace/Thickening.lean | [] | [
"PseudoMetricSpace",
"Set",
"by_cases",
"trans"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
thickening_closure : thickening δ (closure s) = thickening δ s | by
simp_rw [thickening, infEDist_closure] | theorem | Metric.thickening_closure | Topology.MetricSpace | Mathlib/Topology/MetricSpace/Thickening.lean | [] | [
"closure"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
cthickening_closure : cthickening δ (closure s) = cthickening δ s | by
simp_rw [cthickening, infEDist_closure] | theorem | Metric.cthickening_closure | Topology.MetricSpace | Mathlib/Topology/MetricSpace/Thickening.lean | [] | [
"closure"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
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 | Metric.thickening_eq_empty_iff_of_pos | Topology.MetricSpace | Mathlib/Topology/MetricSpace/Thickening.lean | [] | [] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
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] | lemma | Metric.thickening_nonempty_iff_of_pos | Topology.MetricSpace | Mathlib/Topology/MetricSpace/Thickening.lean | [] | [] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
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] | lemma | Metric.thickening_eq_empty_iff | Topology.MetricSpace | Mathlib/Topology/MetricSpace/Thickening.lean | [] | [
"lt_or_ge"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
thickening_nonempty_iff : (thickening ε s).Nonempty ↔ 0 < ε ∧ s.Nonempty | by
simp [nonempty_iff_ne_empty] | lemma | Metric.thickening_nonempty_iff | Topology.MetricSpace | Mathlib/Topology/MetricSpace/Thickening.lean | [] | [] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
_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⟩ ... | theorem | Disjoint.exists_thickenings | Topology.MetricSpace | Mathlib/Topology/MetricSpace/Thickening.lean | [] | [
"Disjoint",
"ENNReal.coe_add",
"ENNReal.ofReal_coe_nnreal",
"IsClosed",
"IsCompact",
"NNReal.coe_div",
"NNReal.coe_two",
"add_halves",
"disjoint_iff_inf_le",
"edist_triangle_left",
"half_pos"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
_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 | Disjoint.exists_cthickenings | Topology.MetricSpace | Mathlib/Topology/MetricSpace/Thickening.lean | [] | [
"Disjoint",
"IsClosed",
"IsCompact",
"half_pos"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
_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 | IsCompact.exists_cthickening_subset_open | Topology.MetricSpace | Mathlib/Topology/MetricSpace/Thickening.lean | [] | [
"IsCompact",
"IsOpen"
] | If `s` is compact, `t` is open and `s ⊆ t`, some `cthickening` of `s` is contained in `t`. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
_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 | IsCompact.exists_isCompact_cthickening | Topology.MetricSpace | Mathlib/Topology/MetricSpace/Thickening.lean | [] | [
"IsCompact",
"LocallyCompactSpace",
"exists_compact_superset",
"interior_subset",
"isOpen_interior"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
_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 | IsCompact.exists_thickening_subset_open | Topology.MetricSpace | Mathlib/Topology/MetricSpace/Thickening.lean | [] | [
"IsCompact",
"IsOpen",
"trans"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
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 | Metric.hasBasis_nhdsSet_thickening | Topology.MetricSpace | Mathlib/Topology/MetricSpace/Thickening.lean | [] | [
"IsCompact",
"Set",
"hasBasis_nhdsSet"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
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 | Metric.hasBasis_nhdsSet_cthickening | Topology.MetricSpace | Mathlib/Topology/MetricSpace/Thickening.lean | [] | [
"IsCompact",
"Set",
"hasBasis_nhdsSet"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
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)... | theorem | Metric.cthickening_eq_iInter_cthickening' | Topology.MetricSpace | Mathlib/Topology/MetricSpace/Thickening.lean | [] | [
"ENNReal.coe_nnreal_eq",
"ENNReal.le_of_forall_pos_le_add",
"ENNReal.ofReal_add_le",
"ENNReal.ofReal_le_ofReal",
"Set",
"le_of_lt",
"trans"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
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 | Metric.cthickening_eq_iInter_cthickening | Topology.MetricSpace | Mathlib/Topology/MetricSpace/Thickening.lean | [] | [
"Set"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
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