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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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 _ => thicken... | theorem | Metric.cthickening_eq_iInter_thickening' | Topology.MetricSpace | Mathlib/Topology/MetricSpace/Thickening.lean | [] | [
"Set",
"antisymm",
"lt_of_le_of_lt"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
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 | Metric.cthickening_eq_iInter_thickening | Topology.MetricSpace | Mathlib/Topology/MetricSpace/Thickening.lean | [] | [
"Set"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
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 | Metric.cthickening_eq_iInter_thickening'' | Topology.MetricSpace | Mathlib/Topology/MetricSpace/Thickening.lean | [] | [
"Set"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
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_nonp... | theorem | Metric.closure_eq_iInter_cthickening' | Topology.MetricSpace | Mathlib/Topology/MetricSpace/Thickening.lean | [] | [
"Set",
"by_cases",
"closure",
"not_lt"
] | The closure of a set equals the intersection of its closed thickenings of positive radii
accumulating at zero. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
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 | Metric.closure_eq_iInter_cthickening | Topology.MetricSpace | Mathlib/Topology/MetricSpace/Thickening.lean | [] | [
"Set",
"closure"
] | The closure of a set equals the intersection of its closed thickenings of positive radii. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
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 | Metric.closure_eq_iInter_thickening' | Topology.MetricSpace | Mathlib/Topology/MetricSpace/Thickening.lean | [] | [
"Set",
"closure",
"le_rfl"
] | The closure of a set equals the intersection of its open thickenings of positive radii
accumulating at zero. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
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 | Metric.closure_eq_iInter_thickening | Topology.MetricSpace | Mathlib/Topology/MetricSpace/Thickening.lean | [] | [
"Set",
"closure"
] | The closure of a set equals the intersection of its (open) thickenings of positive radii. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
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 | Metric.frontier_cthickening_subset | Topology.MetricSpace | Mathlib/Topology/MetricSpace/Thickening.lean | [] | [
"ENNReal.ofReal",
"Set",
"continuous_const",
"frontier",
"frontier_le_subset_eq"
] | The frontier of the closed thickening of a set is contained in an `Metric.infEDist` level
set. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
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 | Metric.closedBall_subset_cthickening | Topology.MetricSpace | Mathlib/Topology/MetricSpace/Thickening.lean | [] | [
"PseudoMetricSpace",
"Set",
"trans"
] | The closed ball of radius `δ` centered at a point of `E` is included in the closed
thickening of `E`. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
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 | Metric.cthickening_subset_iUnion_closedBall_of_lt | Topology.MetricSpace | Mathlib/Topology/MetricSpace/Thickening.lean | [] | [
"PseudoMetricSpace",
"Set",
"trans"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
_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... | theorem | IsCompact.cthickening_eq_biUnion_closedBall | Topology.MetricSpace | Mathlib/Topology/MetricSpace/Thickening.lean | [] | [
"ENNReal.ofReal",
"ENNReal.ofReal_le_ofReal_iff",
"IsCompact",
"PseudoMetricSpace",
"Set",
"edist_dist",
"le_of_eq",
"trans"
] | 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`. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
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) = dis... | theorem | Metric.cthickening_eq_biUnion_closedBall | Topology.MetricSpace | Mathlib/Topology/MetricSpace/Thickening.lean | [] | [
"ENNReal.ofReal",
"ENNReal.ofReal_le_ofReal_iff",
"ENNReal.ofReal_toReal_le",
"ProperSpace",
"PseudoMetricSpace",
"Set",
"closure",
"closure_empty",
"trans"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
_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 | IsClosed.cthickening_eq_biUnion_closedBall | Topology.MetricSpace | Mathlib/Topology/MetricSpace/Thickening.lean | [] | [
"IsClosed",
"ProperSpace",
"PseudoMetricSpace",
"Set"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
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_... | theorem | Metric.infEDist_le_infEDist_cthickening_add | Topology.MetricSpace | Mathlib/Topology/MetricSpace/Thickening.lean | [] | [
"ENNReal.add_lt_add_of_lt_of_le",
"ENNReal.ofReal",
"ENNReal.ofReal_lt_top",
"lt_tsub_iff_right",
"tsub_add_cancel_of_le"
] | For the equality, see `infEDist_cthickening`. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
infEDist_le_infEDist_thickening_add :
infEDist x s ≤ infEDist x (thickening δ s) + ENNReal.ofReal δ | infEDist_le_infEDist_cthickening_add.trans <| by gcongr; exact thickening_subset_cthickening ..
@[deprecated (since := "2026-01-08")]
alias infEdist_le_infEdist_thickening_add := infEDist_le_infEDist_thickening_add | theorem | Metric.infEDist_le_infEDist_thickening_add | Topology.MetricSpace | Mathlib/Topology/MetricSpace/Thickening.lean | [] | [
"ENNReal.ofReal"
] | For the equality, see `infEDist_thickening`. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
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... | theorem | Metric.thickening_thickening_subset | Topology.MetricSpace | Mathlib/Topology/MetricSpace/Thickening.lean | [] | [
"ENNReal.add_lt_add",
"ENNReal.ofReal_add",
"Set",
"le_total"
] | For the equality, see `thickening_thickening`. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
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_... | theorem | Metric.thickening_cthickening_subset | Topology.MetricSpace | Mathlib/Topology/MetricSpace/Thickening.lean | [] | [
"ENNReal.add_lt_add_of_lt_of_le",
"ENNReal.ofReal_add",
"ENNReal.ofReal_lt_top",
"Set",
"le_total"
] | For the equality, see `thickening_cthickening`. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
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 (by grw [hx]) | theorem | Metric.cthickening_thickening_subset | Topology.MetricSpace | Mathlib/Topology/MetricSpace/Thickening.lean | [] | [
"ENNReal.ofReal_add",
"Set",
"le_total"
] | For the equality, see `cthickening_thickening`. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
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 (by grw [hx]) | theorem | Metric.cthickening_cthickening_subset | Topology.MetricSpace | Mathlib/Topology/MetricSpace/Thickening.lean | [] | [
"ENNReal.ofReal_add",
"Set"
] | For the equality, see `cthickening_cthickening`. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
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 | Metric.frontier_cthickening_disjoint | Topology.MetricSpace | Mathlib/Topology/MetricSpace/Thickening.lean | [] | [
"Disjoint",
"Pairwise",
"Set",
"frontier"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
thickening_ball (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 | Metric.thickening_ball | Topology.MetricSpace | Mathlib/Topology/MetricSpace/Thickening.lean | [] | [
"thickening_ball"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
tendsto_nhdsSet {f : β → α} (hs₁ : IsCompact s) (hs₂ : Set.Nonempty s) :
Tendsto f l (𝓝ˢ s) ↔ ∀ ε > 0, ∀ᶠ x in l, infDist (f x) s < ε | by
rw [(hasBasis_nhdsSet_thickening hs₁).tendsto_right_iff]
congrm (∀ ε hε, ?_)
simp [mem_thickening_iff_infDist_lt hs₂] | theorem | Metric.tendsto_nhdsSet | Topology.MetricSpace | Mathlib/Topology/MetricSpace/Thickening.lean | [] | [
"IsCompact",
"Set.Nonempty"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
mem_nhdsSet_iff {t : Set α} (hs : IsCompact s) :
t ∈ 𝓝ˢ s ↔ ∃ ε > 0, Metric.thickening ε s ⊆ t | by
rw [(hasBasis_nhdsSet_thickening hs).mem_iff] | theorem | Metric.mem_nhdsSet_iff | Topology.MetricSpace | Mathlib/Topology/MetricSpace/Thickening.lean | [] | [
"IsCompact",
"Metric.thickening",
"Set"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
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.iInter₂_subset δ hδ |>.trans_eq hs | lemma | IsClopen.of_thickening_subset_self | Topology.MetricSpace | Mathlib/Topology/MetricSpace/Thickening.lean | [] | [
"IsClopen",
"Set.iInter₂_subset",
"closure_subset_iff_isClosed",
"le_antisymm"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
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 | IsClopen.of_cthickening_subset_self | Topology.MetricSpace | Mathlib/Topology/MetricSpace/Thickening.lean | [] | [
"IsClopen",
"trans"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
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⟩
... | theorem | IsCompact.exists_thickening_image_subset | Topology.MetricSpace | Mathlib/Topology/MetricSpace/Thickening.lean | [] | [
"ContinuousAt",
"IsCompact",
"IsOpen",
"PseudoEMetricSpace",
"Set",
"inter_mem_nhdsWithin",
"interior",
"isCompact_singleton",
"nhdsSet_mono",
"subset_interior_iff_mem_nhdsSet",
"union_mem_nhdsSet"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
dist (e : α ≃ β) [Dist β] : Dist α | ⟨fun x y ↦ dist (e x) (e y)⟩ | abbrev | Equiv.dist | Topology.MetricSpace | Mathlib/Topology/MetricSpace/TransferInstance.lean | [] | [
"Dist"
] | Transfer a `Dist` across an `Equiv` | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
pseudometricSpace [PseudoMetricSpace β] (e : α ≃ β) : PseudoMetricSpace α | .induced e ‹_› | abbrev | Equiv.pseudometricSpace | Topology.MetricSpace | Mathlib/Topology/MetricSpace/TransferInstance.lean | [] | [
"PseudoMetricSpace"
] | Transfer a `PseudoMetricSpace` across an `Equiv` | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
metricSpace [MetricSpace β] (e : α ≃ β) : MetricSpace α | .induced e e.injective ‹_› | abbrev | Equiv.metricSpace | Topology.MetricSpace | Mathlib/Topology/MetricSpace/TransferInstance.lean | [] | [
"MetricSpace"
] | Transfer a `MetricSpace` across an `Equiv` | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
edist_def (f g : α →ᵤ β) :
edist f g = ⨆ x, edist (toFun f x) (toFun g x) | rfl | lemma | UniformFun.edist_def | Topology.MetricSpace | Mathlib/Topology/MetricSpace/UniformConvergence.lean | [] | [] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
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 | UniformFun.edist_le | Topology.MetricSpace | Mathlib/Topology/MetricSpace/UniformConvergence.lean | [] | [
"iSup_le_iff"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
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 | UniformFun.lipschitzWith_iff | Topology.MetricSpace | Mathlib/Topology/MetricSpace/UniformConvergence.lean | [] | [
"LipschitzWith"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
lipschitzWith_ofFun_iff {f : γ → α → β} {K : ℝ≥0} :
LipschitzWith K (fun x ↦ ofFun (f x)) ↔ ∀ c, LipschitzWith K (f · c) | lipschitzWith_iff | lemma | UniformFun.lipschitzWith_ofFun_iff | Topology.MetricSpace | Mathlib/Topology/MetricSpace/UniformConvergence.lean | [] | [
"LipschitzWith"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
_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 | LipschitzWith.uniformEquicontinuous | Topology.MetricSpace | Mathlib/Topology/MetricSpace/UniformConvergence.lean | [] | [
"LipschitzWith",
"UniformEquicontinuous",
"uniformEquicontinuous_iff_uniformContinuous"
] | 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. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
lipschitzOnWith_iff {f : γ → α →ᵤ β} {K : ℝ≥0} {s : Set γ} :
LipschitzOnWith K f s ↔ ∀ c, LipschitzOnWith K (fun x ↦ toFun (f x) c) s | by
simp [lipschitzOnWith_iff_restrict, lipschitzWith_iff]
rfl | lemma | UniformFun.lipschitzOnWith_iff | Topology.MetricSpace | Mathlib/Topology/MetricSpace/UniformConvergence.lean | [] | [
"LipschitzOnWith",
"Set",
"lipschitzOnWith_iff_restrict"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
lipschitzOnWith_ofFun_iff {f : γ → α → β} {K : ℝ≥0} {s : Set γ} :
LipschitzOnWith K (fun x ↦ ofFun (f x)) s ↔ ∀ c, LipschitzOnWith K (f · c) s | lipschitzOnWith_iff | lemma | UniformFun.lipschitzOnWith_ofFun_iff | Topology.MetricSpace | Mathlib/Topology/MetricSpace/UniformConvergence.lean | [] | [
"LipschitzOnWith",
"Set"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
_root_.LipschitzOnWith.uniformEquicontinuousOn (f : α → γ → β) (K : ℝ≥0) {s : Set γ}
(h : ∀ c, LipschitzOnWith K (f c) s) : UniformEquicontinuousOn f s | by
rw [uniformEquicontinuousOn_iff_uniformContinuousOn]
rw [← lipschitzOnWith_ofFun_iff] at h
exact h.uniformContinuousOn | lemma | LipschitzOnWith.uniformEquicontinuousOn | Topology.MetricSpace | Mathlib/Topology/MetricSpace/UniformConvergence.lean | [] | [
"LipschitzOnWith",
"Set",
"UniformEquicontinuousOn",
"uniformEquicontinuousOn_iff_uniformContinuousOn"
] | If `f : α → γ → β` is a family of a functions, all of which are Lipschitz on `s` with the
same constant, then the family is uniformly equicontinuous on `s`. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
edist_eval_le {f g : α →ᵤ β} {x : α} :
edist (toFun f x) (toFun g x) ≤ edist f g | edist_le.mp le_rfl x | lemma | UniformFun.edist_eval_le | Topology.MetricSpace | Mathlib/Topology/MetricSpace/UniformConvergence.lean | [] | [
"le_rfl"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
lipschitzWith_eval (x : α) :
LipschitzWith 1 (fun f : α →ᵤ β ↦ toFun f x) | by
intro f g
simpa using edist_eval_le | lemma | UniformFun.lipschitzWith_eval | Topology.MetricSpace | Mathlib/Topology/MetricSpace/UniformConvergence.lean | [] | [
"LipschitzWith"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
dist_def [BoundedSpace β] (f g : α →ᵤ β) :
dist f g = ⨆ x, dist (toFun f x) (toFun g x) | rfl | lemma | UniformFun.dist_def | Topology.MetricSpace | Mathlib/Topology/MetricSpace/UniformConvergence.lean | [] | [
"BoundedSpace"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
dist_le [BoundedSpace β] {f g : α →ᵤ β} {C : ℝ} (hC : 0 ≤ C) :
dist f g ≤ C ↔ ∀ x, dist (toFun f x) (toFun g x) ≤ C | by
simp_rw [dist_edist, ← ENNReal.le_ofReal_iff_toReal_le (edist_ne_top _ _) hC, edist_le] | lemma | UniformFun.dist_le | Topology.MetricSpace | Mathlib/Topology/MetricSpace/UniformConvergence.lean | [] | [
"BoundedSpace",
"ENNReal.le_ofReal_iff_toReal_le",
"dist_edist",
"edist_ne_top"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
isometry_ofFun_boundedContinuousFunction [TopologicalSpace α] :
Isometry (ofFun ∘ DFunLike.coe : (α →ᵇ β) → α →ᵤ β) | by
simp [Isometry, edist_def, edist_eq_iSup] | lemma | UniformFun.isometry_ofFun_boundedContinuousFunction | Topology.MetricSpace | Mathlib/Topology/MetricSpace/UniformConvergence.lean | [] | [
"Isometry",
"TopologicalSpace"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
isometry_ofFun_continuousMap [TopologicalSpace α] [CompactSpace α] :
Isometry (ofFun ∘ DFunLike.coe : C(α, β) → α →ᵤ β) | isometry_ofFun_boundedContinuousFunction.comp <|
ContinuousMap.isometryEquivBoundedOfCompact α β |>.isometry | lemma | UniformFun.isometry_ofFun_continuousMap | Topology.MetricSpace | Mathlib/Topology/MetricSpace/UniformConvergence.lean | [] | [
"CompactSpace",
"ContinuousMap.isometryEquivBoundedOfCompact",
"Isometry",
"TopologicalSpace"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
edist_continuousMapMk [TopologicalSpace α] [CompactSpace α]
{f g : α →ᵤ β} (hf : Continuous (toFun f)) (hg : Continuous (toFun g)) :
edist (⟨_, hf⟩ : C(α, β)) ⟨_, hg⟩ = edist f g | by
simp [← isometry_ofFun_continuousMap.edist_eq] | lemma | UniformFun.edist_continuousMapMk | Topology.MetricSpace | Mathlib/Topology/MetricSpace/UniformConvergence.lean | [] | [
"CompactSpace",
"Continuous",
"TopologicalSpace"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
continuous_of_forall_lipschitzWith {f : γ → α →ᵤ[𝔖] β} (K : Set α → ℝ≥0)
(h : ∀ s ∈ 𝔖, ∀ c ∈ s, LipschitzWith (K s) (fun x ↦ toFun 𝔖 (f x) c)) :
Continuous f | by
rw [UniformOnFun.continuous_rng_iff]
refine fun s hs ↦ LipschitzWith.continuous (K := K s) ?_
rw [UniformFun.lipschitzWith_iff]
rintro ⟨y, hy⟩
exact h s hs y hy | lemma | UniformOnFun.continuous_of_forall_lipschitzWith | Topology.MetricSpace | Mathlib/Topology/MetricSpace/UniformConvergence.lean | [] | [
"Continuous",
"LipschitzWith",
"LipschitzWith.continuous",
"Set",
"UniformFun.lipschitzWith_iff",
"UniformOnFun.continuous_rng_iff"
] | Let `f : γ → α →ᵤ[𝔖] β`. If for every `s ∈ 𝔖` and for every `c ∈ s`, the function
`fun x ↦ f x c` is Lipschitz (with Lipschitz constant depending on `s`), then `f` is continuous. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
edist_def [Finite 𝔖] (f g : α →ᵤ[𝔖] β) :
edist f g = ⨆ x ∈ ⋃₀ 𝔖, edist (toFun 𝔖 f x) (toFun 𝔖 g x) | rfl | lemma | UniformOnFun.edist_def | Topology.MetricSpace | Mathlib/Topology/MetricSpace/UniformConvergence.lean | [] | [
"Finite"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
edist_def' [Finite 𝔖] (f g : α →ᵤ[𝔖] β) :
edist f g = ⨆ s ∈ 𝔖, ⨆ x ∈ s, edist (toFun 𝔖 f x) (toFun 𝔖 g x) | by
simp [edist_def, iSup_and, iSup_comm (ι := α)] | lemma | UniformOnFun.edist_def' | Topology.MetricSpace | Mathlib/Topology/MetricSpace/UniformConvergence.lean | [] | [
"Finite",
"iSup_and",
"iSup_comm"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
edist_eq_restrict_sUnion [Finite 𝔖] {f g : α →ᵤ[𝔖] β} :
edist f g = edist
(UniformFun.ofFun ((⋃₀ 𝔖).restrict (toFun 𝔖 f)))
(UniformFun.ofFun ((⋃₀ 𝔖).restrict (toFun 𝔖 g))) | iSup_subtype' | lemma | UniformOnFun.edist_eq_restrict_sUnion | Topology.MetricSpace | Mathlib/Topology/MetricSpace/UniformConvergence.lean | [] | [
"Finite",
"UniformFun.ofFun",
"iSup_subtype'"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
edist_eq_pi_restrict [Fintype 𝔖] {f g : α →ᵤ[𝔖] β} :
edist f g = edist
(fun s : 𝔖 ↦ UniformFun.ofFun ((s : Set α).restrict (toFun 𝔖 f)))
(fun s : 𝔖 ↦ UniformFun.ofFun ((s : Set α).restrict (toFun 𝔖 g))) | by
simp_rw [edist_def', iSup_subtype', edist_pi_def, Finset.sup_univ_eq_iSup]
rfl | lemma | UniformOnFun.edist_eq_pi_restrict | Topology.MetricSpace | Mathlib/Topology/MetricSpace/UniformConvergence.lean | [] | [
"Finset.sup_univ_eq_iSup",
"Fintype",
"Set",
"UniformFun.ofFun",
"edist_pi_def",
"iSup_subtype'"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
edist_le {f g : α →ᵤ[𝔖] β} {C : ℝ≥0∞} :
edist f g ≤ C ↔ ∀ x ∈ ⋃₀ 𝔖, edist (toFun 𝔖 f x) (toFun 𝔖 g x) ≤ C | by
simp_rw [edist_def, iSup₂_le_iff] | lemma | UniformOnFun.edist_le | Topology.MetricSpace | Mathlib/Topology/MetricSpace/UniformConvergence.lean | [] | [
"iSup₂_le_iff"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
lipschitzWith_iff {f : γ → α →ᵤ[𝔖] β} {K : ℝ≥0} :
LipschitzWith K f ↔ ∀ c ∈ ⋃₀ 𝔖, LipschitzWith K (fun x ↦ toFun 𝔖 (f x) c) | by
simp [LipschitzWith, edist_le]
tauto | lemma | UniformOnFun.lipschitzWith_iff | Topology.MetricSpace | Mathlib/Topology/MetricSpace/UniformConvergence.lean | [] | [
"LipschitzWith"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
lipschitzOnWith_iff {f : γ → α →ᵤ[𝔖] β} {K : ℝ≥0} {s : Set γ} :
LipschitzOnWith K f s ↔ ∀ c ∈ ⋃₀ 𝔖, LipschitzOnWith K (fun x ↦ toFun 𝔖 (f x) c) s | by
simp [lipschitzOnWith_iff_restrict, lipschitzWith_iff]
rfl | lemma | UniformOnFun.lipschitzOnWith_iff | Topology.MetricSpace | Mathlib/Topology/MetricSpace/UniformConvergence.lean | [] | [
"LipschitzOnWith",
"Set",
"lipschitzOnWith_iff_restrict"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
edist_eval_le {f g : α →ᵤ[𝔖] β} {x : α} (hx : x ∈ ⋃₀ 𝔖) :
edist (toFun 𝔖 f x) (toFun 𝔖 g x) ≤ edist f g | edist_le.mp le_rfl x hx | lemma | UniformOnFun.edist_eval_le | Topology.MetricSpace | Mathlib/Topology/MetricSpace/UniformConvergence.lean | [] | [
"le_rfl"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
lipschitzWith_eval {x : α} (hx : x ∈ ⋃₀ 𝔖) :
LipschitzWith 1 (fun f : α →ᵤ[𝔖] β ↦ toFun 𝔖 f x) | by
intro f g
simpa only [ENNReal.coe_one, one_mul] using edist_eval_le hx | lemma | UniformOnFun.lipschitzWith_eval | Topology.MetricSpace | Mathlib/Topology/MetricSpace/UniformConvergence.lean | [] | [
"ENNReal.coe_one",
"LipschitzWith",
"one_mul"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
lipschitzWith_one_ofFun_toFun :
LipschitzWith 1 (ofFun 𝔖 ∘ UniformFun.toFun : (α →ᵤ β) → (α →ᵤ[𝔖] β)) | lipschitzWith_iff.mpr fun _ _ ↦ UniformFun.lipschitzWith_eval _ | lemma | UniformOnFun.lipschitzWith_one_ofFun_toFun | Topology.MetricSpace | Mathlib/Topology/MetricSpace/UniformConvergence.lean | [] | [
"LipschitzWith",
"UniformFun.lipschitzWith_eval",
"UniformFun.toFun"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
lipschitzWith_one_ofFun_toFun' [Finite 𝔗] (h : ⋃₀ 𝔖 ⊆ ⋃₀ 𝔗) :
LipschitzWith 1 (ofFun 𝔖 ∘ toFun 𝔗 : (α →ᵤ[𝔗] β) → (α →ᵤ[𝔖] β)) | lipschitzWith_iff.mpr fun _x hx ↦ lipschitzWith_eval (h hx) | lemma | UniformOnFun.lipschitzWith_one_ofFun_toFun' | Topology.MetricSpace | Mathlib/Topology/MetricSpace/UniformConvergence.lean | [] | [
"Finite",
"LipschitzWith"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
lipschitzWith_restrict (s : Set α) (hs : s ∈ 𝔖) :
LipschitzWith 1 (UniformFun.ofFun ∘ s.restrict ∘ toFun 𝔖 : (α →ᵤ[𝔖] β) → (s →ᵤ β)) | UniformFun.lipschitzWith_iff.mpr fun x ↦ lipschitzWith_eval ⟨s, hs, x.2⟩ | lemma | UniformOnFun.lipschitzWith_restrict | Topology.MetricSpace | Mathlib/Topology/MetricSpace/UniformConvergence.lean | [] | [
"LipschitzWith",
"Set",
"UniformFun.ofFun"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
isometry_restrict (s : Set α) :
Isometry (UniformFun.ofFun ∘ s.restrict ∘ toFun {s} : (α →ᵤ[{s}] β) → (s →ᵤ β)) | by
simp [Isometry, edist_def, UniformFun.edist_def, iSup_subtype] | lemma | UniformOnFun.isometry_restrict | Topology.MetricSpace | Mathlib/Topology/MetricSpace/UniformConvergence.lean | [] | [
"Isometry",
"Set",
"UniformFun.edist_def",
"UniformFun.ofFun",
"iSup_subtype"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
edist_continuousRestrict [TopologicalSpace α] {f g : α →ᵤ[𝔖] β}
[CompactSpace (⋃₀ 𝔖)] (hf : ContinuousOn (toFun 𝔖 f) (⋃₀ 𝔖))
(hg : ContinuousOn (toFun 𝔖 g) (⋃₀ 𝔖)) :
edist (⟨_, hf.restrict⟩ : C(⋃₀ 𝔖, β)) ⟨_, hg.restrict⟩ = edist f g | by
simp [ContinuousMap.edist_eq_iSup, iSup_subtype, edist_def] | lemma | UniformOnFun.edist_continuousRestrict | Topology.MetricSpace | Mathlib/Topology/MetricSpace/UniformConvergence.lean | [] | [
"CompactSpace",
"ContinuousMap.edist_eq_iSup",
"ContinuousOn",
"TopologicalSpace",
"iSup_subtype"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
edist_continuousRestrict_of_singleton [TopologicalSpace α] {s : Set α}
{f g : α →ᵤ[{s}] β} [CompactSpace s] (hf : ContinuousOn (toFun {s} f) s)
(hg : ContinuousOn (toFun {s} g) s) :
edist (⟨_, hf.restrict⟩ : C(s, β)) ⟨_, hg.restrict⟩ = edist f g | by
simp [ContinuousMap.edist_eq_iSup, iSup_subtype, edist_def] | lemma | UniformOnFun.edist_continuousRestrict_of_singleton | Topology.MetricSpace | Mathlib/Topology/MetricSpace/UniformConvergence.lean | [] | [
"CompactSpace",
"ContinuousMap.edist_eq_iSup",
"ContinuousOn",
"Set",
"TopologicalSpace",
"iSup_subtype"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
exists_pos_lt_subset_ball (hr : 0 < r) (hs : IsClosed s) (h : s ⊆ ball x r) :
∃ r' ∈ Ioo 0 r, s ⊆ ball x r' | by
rcases eq_empty_or_nonempty s with (rfl | hne)
· exact ⟨r / 2, ⟨half_pos hr, half_lt_self hr⟩, empty_subset _⟩
have : IsCompact s :=
(isCompact_closedBall x r).of_isClosed_subset hs (h.trans ball_subset_closedBall)
obtain ⟨y, hys, hy⟩ : ∃ y ∈ s, s ⊆ closedBall x (dist y x) :=
this.exists_isMaxOn (β :... | theorem | exists_pos_lt_subset_ball | Topology.MetricSpace.ProperSpace | Mathlib/Topology/MetricSpace/ProperSpace/Lemmas.lean | [] | [
"IsClosed",
"IsCompact",
"exists_between"
] | If a nonempty ball in a proper space includes a closed set `s`, then there exists a nonempty
ball with the same center and a strictly smaller radius that includes `s`. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
exists_lt_subset_ball (hs : IsClosed s) (h : s ⊆ ball x r) : ∃ r' < r, s ⊆ ball x r' | by
rcases le_or_gt r 0 with hr | hr
· rw [ball_eq_empty.2 hr, subset_empty_iff] at h
subst s
exact (exists_lt r).imp fun r' hr' => ⟨hr', empty_subset _⟩
· exact (exists_pos_lt_subset_ball hr hs h).imp fun r' hr' => ⟨hr'.1.2, hr'.2⟩ | theorem | exists_lt_subset_ball | Topology.MetricSpace.ProperSpace | Mathlib/Topology/MetricSpace/ProperSpace/Lemmas.lean | [] | [
"IsClosed",
"exists_pos_lt_subset_ball",
"le_or_gt"
] | If a ball in a proper space includes a closed set `s`, then there exists a ball with the same
center and a strictly smaller radius that includes `s`. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
Metric.exists_isLocalMin_mem_ball [TopologicalSpace β]
[ConditionallyCompleteLinearOrder β] [OrderTopology β] {f : α → β} {a z : α} {r : ℝ}
(hf : ContinuousOn f (closedBall a r)) (hz : z ∈ closedBall a r)
(hf1 : ∀ z' ∈ sphere a r, f z < f z') : ∃ z ∈ ball a r, IsLocalMin f z | by
simp_rw [← closedBall_diff_ball] at hf1
exact (isCompact_closedBall a r).exists_isLocalMin_mem_open ball_subset_closedBall hf hz hf1
isOpen_ball | theorem | Metric.exists_isLocalMin_mem_ball | Topology.MetricSpace.ProperSpace | Mathlib/Topology/MetricSpace/ProperSpace/Lemmas.lean | [] | [
"ConditionallyCompleteLinearOrder",
"ContinuousOn",
"IsLocalMin",
"OrderTopology",
"TopologicalSpace"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
isProperMap_dist (x : α) : IsProperMap (dist x) | isProperMap_iff_tendsto_cocompact.mpr
⟨by fun_prop, (tendsto_dist_left_cocompact_atTop x).trans atTop_le_cocompact⟩ | lemma | isProperMap_dist | Topology.MetricSpace.ProperSpace | Mathlib/Topology/MetricSpace/ProperSpace/Lemmas.lean | [] | [
"IsProperMap",
"ProperSpace",
"tendsto_dist_left_cocompact_atTop",
"trans"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
properSpace_iff_isProperMap_dist : ProperSpace α ↔ ∀ x : α, IsProperMap (dist x) | by
refine ⟨fun _ ↦ isProperMap_dist, fun H ↦ ⟨fun x r ↦ ?_⟩⟩
convert! (H x).isCompact_preimage (isCompact_closedBall 0 r)
ext
simp [dist_comm, Real.dist_eq] | lemma | properSpace_iff_isProperMap_dist | Topology.MetricSpace.ProperSpace | Mathlib/Topology/MetricSpace/ProperSpace/Lemmas.lean | [] | [
"IsProperMap",
"ProperSpace",
"Real.dist_eq",
"dist_comm",
"isProperMap_dist"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
isClosedMap_dist (x : α) : IsClosedMap (dist x) | (isProperMap_dist x).isClosedMap | lemma | isClosedMap_dist | Topology.MetricSpace.ProperSpace | Mathlib/Topology/MetricSpace/ProperSpace/Lemmas.lean | [] | [
"IsClosedMap",
"isProperMap_dist"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
isProperMap_nndist (x : α) : IsProperMap (nndist x) | isProperMap_of_comp_of_inj (Z := ℝ) (g := (↑)) (by fun_prop) (by fun_prop)
(isProperMap_dist x) NNReal.coe_injective | lemma | isProperMap_nndist | Topology.MetricSpace.ProperSpace | Mathlib/Topology/MetricSpace/ProperSpace/Lemmas.lean | [] | [
"IsProperMap",
"NNReal.coe_injective",
"isProperMap_dist",
"isProperMap_of_comp_of_inj"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
isClosedMap_nndist (x : α) : IsClosedMap (nndist x) | (isProperMap_nndist _).isClosedMap | lemma | isClosedMap_nndist | Topology.MetricSpace.ProperSpace | Mathlib/Topology/MetricSpace/ProperSpace/Lemmas.lean | [] | [
"IsClosedMap",
"isProperMap_nndist"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
instProperSpace : ProperSpace ℝ≥0 | where
isCompact_closedBall x r := by
have emb : IsClosedEmbedding ((↑) : ℝ≥0 → ℝ) := Isometry.isClosedEmbedding fun _ ↦ congrFun rfl
exact emb.isCompact_preimage (K := Metric.closedBall x r) (isCompact_closedBall _ _) | instance | NNReal.instProperSpace | Topology.MetricSpace.ProperSpace | Mathlib/Topology/MetricSpace/ProperSpace/Real.lean | [] | [
"Isometry.isClosedEmbedding",
"Metric.closedBall",
"ProperSpace"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
dist_le_Ico_sum_dist (f : ℕ → α) {m n} (h : m ≤ n) :
dist (f m) (f n) ≤ ∑ i ∈ Finset.Ico m n, dist (f i) (f (i + 1)) | by
induction n, h using Nat.le_induction with
| base => rw [Finset.Ico_self, Finset.sum_empty, dist_self]
| succ n hle ihn =>
calc
dist (f m) (f (n + 1)) ≤ dist (f m) (f n) + dist (f n) (f (n + 1)) := dist_triangle _ _ _
_ ≤ (∑ i ∈ Finset.Ico m n, _) + _ := add_le_add ihn le_rfl
_ = ∑ i ∈ Fi... | theorem | dist_le_Ico_sum_dist | Topology.MetricSpace.Pseudo | Mathlib/Topology/MetricSpace/Pseudo/Basic.lean | [] | [
"Finset.Ico",
"Finset.Ico_self",
"Finset.insert_Ico_right_eq_Ico_add_one",
"Nat.le_induction",
"dist_self",
"dist_triangle",
"le_rfl"
] | The triangle (polygon) inequality for sequences of points; `Finset.Ico` version. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
dist_le_range_sum_dist (f : ℕ → α) (n : ℕ) :
dist (f 0) (f n) ≤ ∑ i ∈ Finset.range n, dist (f i) (f (i + 1)) | Nat.Ico_zero_eq_range n ▸ dist_le_Ico_sum_dist f (Nat.zero_le n) | theorem | dist_le_range_sum_dist | Topology.MetricSpace.Pseudo | Mathlib/Topology/MetricSpace/Pseudo/Basic.lean | [] | [
"Finset.range",
"Nat.Ico_zero_eq_range",
"dist_le_Ico_sum_dist"
] | The triangle (polygon) inequality for sequences of points; `Finset.range` version. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
dist_le_Ico_sum_of_dist_le {f : ℕ → α} {m n} (hmn : m ≤ n) {d : ℕ → ℝ}
(hd : ∀ {k}, m ≤ k → k < n → dist (f k) (f (k + 1)) ≤ d k) :
dist (f m) (f n) ≤ ∑ i ∈ Finset.Ico m n, d i | le_trans (dist_le_Ico_sum_dist f hmn) <|
Finset.sum_le_sum fun _k hk => hd (Finset.mem_Ico.1 hk).1 (Finset.mem_Ico.1 hk).2 | theorem | dist_le_Ico_sum_of_dist_le | Topology.MetricSpace.Pseudo | Mathlib/Topology/MetricSpace/Pseudo/Basic.lean | [] | [
"Finset.Ico",
"dist_le_Ico_sum_dist",
"le_trans"
] | A version of `dist_le_Ico_sum_dist` with each intermediate distance replaced
with an upper estimate. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
dist_le_range_sum_of_dist_le {f : ℕ → α} (n : ℕ) {d : ℕ → ℝ}
(hd : ∀ {k}, k < n → dist (f k) (f (k + 1)) ≤ d k) :
dist (f 0) (f n) ≤ ∑ i ∈ Finset.range n, d i | Nat.Ico_zero_eq_range n ▸ dist_le_Ico_sum_of_dist_le zero_le fun _ => hd | theorem | dist_le_range_sum_of_dist_le | Topology.MetricSpace.Pseudo | Mathlib/Topology/MetricSpace/Pseudo/Basic.lean | [] | [
"Finset.range",
"Nat.Ico_zero_eq_range",
"dist_le_Ico_sum_of_dist_le"
] | A version of `dist_le_range_sum_dist` with each intermediate distance replaced
with an upper estimate. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
isUniformInducing_iff [PseudoMetricSpace β] {f : α → β} :
IsUniformInducing f ↔ UniformContinuous f ∧
∀ δ > 0, ∃ ε > 0, ∀ {a b : α}, dist (f a) (f b) < ε → dist a b < δ | isUniformInducing_iff'.trans <| Iff.rfl.and <|
((uniformity_basis_dist.comap _).le_basis_iff uniformity_basis_dist).trans <| by
simp only [subset_def, Prod.forall, gt_iff_lt, preimage_setOf_eq, Prod.map_apply, mem_setOf] | theorem | Metric.isUniformInducing_iff | Topology.MetricSpace.Pseudo | Mathlib/Topology/MetricSpace/Pseudo/Basic.lean | [] | [
"IsUniformInducing",
"PseudoMetricSpace",
"UniformContinuous",
"trans"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
isUniformEmbedding_iff [PseudoMetricSpace β] {f : α → β} :
IsUniformEmbedding f ↔ Function.Injective f ∧ UniformContinuous f ∧
∀ δ > 0, ∃ ε > 0, ∀ {a b : α}, dist (f a) (f b) < ε → dist a b < δ | by
rw [isUniformEmbedding_iff, and_comm, isUniformInducing_iff] | theorem | Metric.isUniformEmbedding_iff | Topology.MetricSpace.Pseudo | Mathlib/Topology/MetricSpace/Pseudo/Basic.lean | [] | [
"IsUniformEmbedding",
"PseudoMetricSpace",
"UniformContinuous"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
controlled_of_isUniformInducing [PseudoMetricSpace β] {f : α → β}
(h : IsUniformInducing f) :
(∀ ε > 0, ∃ δ > 0, ∀ {a b : α}, dist a b < δ → dist (f a) (f b) < ε) ∧
∀ δ > 0, ∃ ε > 0, ∀ {a b : α}, dist (f a) (f b) < ε → dist a b < δ | ⟨uniformContinuous_iff.1 h.uniformContinuous, (isUniformInducing_iff.1 h).2⟩ | theorem | Metric.controlled_of_isUniformInducing | Topology.MetricSpace.Pseudo | Mathlib/Topology/MetricSpace/Pseudo/Basic.lean | [] | [
"IsUniformInducing",
"PseudoMetricSpace"
] | If a map between pseudometric spaces is a uniform inducing map then the distance between `f x`
and `f y` is controlled in terms of the distance between `x` and `y`. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
controlled_of_isUniformEmbedding [PseudoMetricSpace β] {f : α → β}
(h : IsUniformEmbedding f) :
(∀ ε > 0, ∃ δ > 0, ∀ {a b : α}, dist a b < δ → dist (f a) (f b) < ε) ∧
∀ δ > 0, ∃ ε > 0, ∀ {a b : α}, dist (f a) (f b) < ε → dist a b < δ | controlled_of_isUniformInducing h.toIsUniformInducing | theorem | Metric.controlled_of_isUniformEmbedding | Topology.MetricSpace.Pseudo | Mathlib/Topology/MetricSpace/Pseudo/Basic.lean | [] | [
"IsUniformEmbedding",
"PseudoMetricSpace"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
totallyBounded_iff {s : Set α} :
TotallyBounded s ↔ ∀ ε > 0, ∃ t : Set α, t.Finite ∧ s ⊆ ⋃ y ∈ t, ball y ε | uniformity_basis_dist.totallyBounded_iff | theorem | Metric.totallyBounded_iff | Topology.MetricSpace.Pseudo | Mathlib/Topology/MetricSpace/Pseudo/Basic.lean | [] | [
"Set",
"TotallyBounded"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
totallyBounded_of_finite_discretization {s : Set α}
(H : ∀ ε > (0 : ℝ),
∃ (β : Type u) (_ : Fintype β) (F : s → β), ∀ x y, F x = F y → dist (x : α) y < ε) :
TotallyBounded s | by
rcases s.eq_empty_or_nonempty with hs | hs
· rw [hs]
exact totallyBounded_empty
rcases hs with ⟨x0, hx0⟩
haveI : Inhabited s := ⟨⟨x0, hx0⟩⟩
refine totallyBounded_iff.2 fun ε ε0 => ?_
rcases H ε ε0 with ⟨β, fβ, F, hF⟩
let Finv := Function.invFun F
refine ⟨range (Subtype.val ∘ Finv), finite_range _... | theorem | Metric.totallyBounded_of_finite_discretization | Topology.MetricSpace.Pseudo | Mathlib/Topology/MetricSpace/Pseudo/Basic.lean | [] | [
"Fintype",
"Function.invFun",
"Function.invFun_eq",
"Set",
"Set.mem_iUnion",
"Set.mem_range",
"TotallyBounded",
"totallyBounded_empty"
] | A pseudometric space is totally bounded if one can reconstruct up to any ε>0 any element of the
space from finitely many data. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
finite_approx_of_totallyBounded {s : Set α} (hs : TotallyBounded s) :
∀ ε > 0, ∃ t, t ⊆ s ∧ Set.Finite t ∧ s ⊆ ⋃ y ∈ t, ball y ε | by
intro ε ε_pos
rw [totallyBounded_iff_subset] at hs
exact hs _ (dist_mem_uniformity ε_pos) | theorem | Metric.finite_approx_of_totallyBounded | Topology.MetricSpace.Pseudo | Mathlib/Topology/MetricSpace/Pseudo/Basic.lean | [] | [
"Set",
"Set.Finite",
"TotallyBounded",
"totallyBounded_iff_subset"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
tendstoUniformlyOnFilter_iff {F : ι → β → α} {f : β → α} {p : Filter ι} {p' : Filter β} :
TendstoUniformlyOnFilter F f p p' ↔
∀ ε > 0, ∀ᶠ n : ι × β in p ×ˢ p', dist (f n.snd) (F n.fst n.snd) < ε | by
refine ⟨fun H ε hε => H _ (dist_mem_uniformity hε), fun H u hu => ?_⟩
rcases mem_uniformity_dist.1 hu with ⟨ε, εpos, hε⟩
exact (H ε εpos).mono fun n hn => hε hn | theorem | Metric.tendstoUniformlyOnFilter_iff | Topology.MetricSpace.Pseudo | Mathlib/Topology/MetricSpace/Pseudo/Basic.lean | [] | [
"Filter",
"TendstoUniformlyOnFilter"
] | Expressing uniform convergence using `dist` | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
tendstoLocallyUniformlyOn_iff [TopologicalSpace β] {F : ι → β → α} {f : β → α}
{p : Filter ι} {s : Set β} :
TendstoLocallyUniformlyOn F f p s ↔
∀ ε > 0, ∀ x ∈ s, ∃ t ∈ 𝓝[s] x, ∀ᶠ n in p, ∀ y ∈ t, dist (f y) (F n y) < ε | by
refine ⟨fun H ε hε => H _ (dist_mem_uniformity hε), fun H u hu x hx => ?_⟩
rcases mem_uniformity_dist.1 hu with ⟨ε, εpos, hε⟩
rcases H ε εpos x hx with ⟨t, ht, Ht⟩
exact ⟨t, ht, Ht.mono fun n hs x hx => hε (hs x hx)⟩ | theorem | Metric.tendstoLocallyUniformlyOn_iff | Topology.MetricSpace.Pseudo | Mathlib/Topology/MetricSpace/Pseudo/Basic.lean | [] | [
"Filter",
"Set",
"TendstoLocallyUniformlyOn",
"TopologicalSpace"
] | Expressing locally uniform convergence on a set using `dist`. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
tendstoUniformlyOn_iff {F : ι → β → α} {f : β → α} {p : Filter ι} {s : Set β} :
TendstoUniformlyOn F f p s ↔ ∀ ε > 0, ∀ᶠ n in p, ∀ x ∈ s, dist (f x) (F n x) < ε | by
refine ⟨fun H ε hε => H _ (dist_mem_uniformity hε), fun H u hu => ?_⟩
rcases mem_uniformity_dist.1 hu with ⟨ε, εpos, hε⟩
exact (H ε εpos).mono fun n hs x hx => hε (hs x hx) | theorem | Metric.tendstoUniformlyOn_iff | Topology.MetricSpace.Pseudo | Mathlib/Topology/MetricSpace/Pseudo/Basic.lean | [] | [
"Filter",
"Set",
"TendstoUniformlyOn"
] | Expressing uniform convergence on a set using `dist`. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
tendstoLocallyUniformly_iff [TopologicalSpace β] {F : ι → β → α} {f : β → α}
{p : Filter ι} :
TendstoLocallyUniformly F f p ↔
∀ ε > 0, ∀ x : β, ∃ t ∈ 𝓝 x, ∀ᶠ n in p, ∀ y ∈ t, dist (f y) (F n y) < ε | by
simp only [← tendstoLocallyUniformlyOn_univ, tendstoLocallyUniformlyOn_iff, nhdsWithin_univ,
mem_univ, forall_const] | theorem | Metric.tendstoLocallyUniformly_iff | Topology.MetricSpace.Pseudo | Mathlib/Topology/MetricSpace/Pseudo/Basic.lean | [] | [
"Filter",
"TendstoLocallyUniformly",
"TopologicalSpace",
"nhdsWithin_univ",
"tendstoLocallyUniformlyOn_univ"
] | Expressing locally uniform convergence using `dist`. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
tendstoUniformly_iff {F : ι → β → α} {f : β → α} {p : Filter ι} :
TendstoUniformly F f p ↔ ∀ ε > 0, ∀ᶠ n in p, ∀ x, dist (f x) (F n x) < ε | by
rw [← tendstoUniformlyOn_univ, tendstoUniformlyOn_iff]
simp | theorem | Metric.tendstoUniformly_iff | Topology.MetricSpace.Pseudo | Mathlib/Topology/MetricSpace/Pseudo/Basic.lean | [] | [
"Filter",
"TendstoUniformly",
"tendstoUniformlyOn_univ"
] | Expressing uniform convergence using `dist`. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
cauchy_iff {f : Filter α} :
Cauchy f ↔ NeBot f ∧ ∀ ε > 0, ∃ t ∈ f, ∀ x ∈ t, ∀ y ∈ t, dist x y < ε | uniformity_basis_dist.cauchy_iff | theorem | Metric.cauchy_iff | Topology.MetricSpace.Pseudo | Mathlib/Topology/MetricSpace/Pseudo/Basic.lean | [] | [
"Cauchy",
"Filter",
"cauchy_iff"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
exists_ball_inter_eq_singleton_of_mem_discrete (hs : IsDiscrete s) {x : α} (hx : x ∈ s) :
∃ ε > 0, Metric.ball x ε ∩ s = {x} | nhds_basis_ball.exists_inter_eq_singleton_of_mem_discrete hs hx | theorem | Metric.exists_ball_inter_eq_singleton_of_mem_discrete | Topology.MetricSpace.Pseudo | Mathlib/Topology/MetricSpace/Pseudo/Basic.lean | [] | [
"IsDiscrete",
"Metric.ball"
] | Given a point `x` in a discrete subset `s` of a pseudometric space, there is an open ball
centered at `x` and intersecting `s` only at `x`. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
exists_closedBall_inter_eq_singleton_of_discrete (hs : IsDiscrete s) {x : α} (hx : x ∈ s) :
∃ ε > 0, Metric.closedBall x ε ∩ s = {x} | nhds_basis_closedBall.exists_inter_eq_singleton_of_mem_discrete hs hx | theorem | Metric.exists_closedBall_inter_eq_singleton_of_discrete | Topology.MetricSpace.Pseudo | Mathlib/Topology/MetricSpace/Pseudo/Basic.lean | [] | [
"IsDiscrete",
"Metric.closedBall"
] | Given a point `x` in a discrete subset `s` of a pseudometric space, there is a closed ball
of positive radius centered at `x` and intersecting `s` only at `x`. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
Metric.inseparable_iff_nndist {x y : α} : Inseparable x y ↔ nndist x y = 0 | by
rw [EMetric.inseparable_iff, edist_nndist, ENNReal.coe_eq_zero]
alias ⟨Inseparable.nndist_eq_zero, _⟩ := Metric.inseparable_iff_nndist | theorem | Metric.inseparable_iff_nndist | Topology.MetricSpace.Pseudo | Mathlib/Topology/MetricSpace/Pseudo/Basic.lean | [] | [
"EMetric.inseparable_iff",
"ENNReal.coe_eq_zero",
"Inseparable",
"edist_nndist"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
Metric.inseparable_iff {x y : α} : Inseparable x y ↔ dist x y = 0 | by
rw [Metric.inseparable_iff_nndist, dist_nndist, NNReal.coe_eq_zero]
alias ⟨Inseparable.dist_eq_zero, _⟩ := Metric.inseparable_iff | theorem | Metric.inseparable_iff | Topology.MetricSpace.Pseudo | Mathlib/Topology/MetricSpace/Pseudo/Basic.lean | [] | [
"Inseparable",
"Metric.inseparable_iff_nndist",
"NNReal.coe_eq_zero",
"dist_nndist"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
tendsto_nhds_unique_dist {f : β → α} {l : Filter β} {x y : α} [NeBot l]
(ha : Tendsto f l (𝓝 x)) (hb : Tendsto f l (𝓝 y)) : dist x y = 0 | (tendsto_nhds_unique_inseparable ha hb).dist_eq_zero | theorem | tendsto_nhds_unique_dist | Topology.MetricSpace.Pseudo | Mathlib/Topology/MetricSpace/Pseudo/Basic.lean | [] | [
"Filter",
"dist_eq_zero",
"tendsto_nhds_unique_inseparable"
] | A weaker version of `tendsto_nhds_unique` for `PseudoMetricSpace`. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
cauchySeq_iff_tendsto_dist_atTop_0 [Nonempty β] [SemilatticeSup β] {u : β → α} :
CauchySeq u ↔ Tendsto (fun n : β × β => dist (u n.1) (u n.2)) atTop (𝓝 0) | by
rw [cauchySeq_iff_tendsto, Metric.uniformity_eq_comap_nhds_zero, tendsto_comap_iff,
Function.comp_def]
simp_rw [Prod.map_fst, Prod.map_snd] | theorem | cauchySeq_iff_tendsto_dist_atTop_0 | Topology.MetricSpace.Pseudo | Mathlib/Topology/MetricSpace/Pseudo/Basic.lean | [] | [
"CauchySeq",
"Metric.uniformity_eq_comap_nhds_zero",
"SemilatticeSup",
"cauchySeq_iff_tendsto"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
IsInducing.isSeparable_preimage {α : Type*} [TopologicalSpace α]
[PseudoMetrizableSpace α] {f : β → α} [TopologicalSpace β]
(hf : IsInducing f) {s : Set α} (hs : IsSeparable s) : IsSeparable (f ⁻¹' s) | by
letI : UniformSpace α := TopologicalSpace.pseudoMetrizableSpaceUniformity α
have := pseudoMetrizableSpaceUniformity_countably_generated
have : SeparableSpace s := hs.separableSpace
have : SecondCountableTopology s := UniformSpace.secondCountable_of_separable _
have : IsInducing ((mapsTo_preimage f s).restr... | lemma | Topology.IsInducing.isSeparable_preimage | Topology.MetricSpace.Pseudo | Mathlib/Topology/MetricSpace/Pseudo/Basic.lean | [] | [
"IsSeparable",
"SecondCountableTopology",
"Set",
"TopologicalSpace",
"TopologicalSpace.pseudoMetrizableSpaceUniformity",
"UniformSpace",
"UniformSpace.secondCountable_of_separable"
] | The preimage of a separable set by an inducing map is separable. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
IsEmbedding.isSeparable_preimage {α : Type*} [TopologicalSpace α]
[PseudoMetrizableSpace α] {f : β → α} [TopologicalSpace β]
(hf : IsEmbedding f) {s : Set α} (hs : IsSeparable s) : IsSeparable (f ⁻¹' s) | hf.isInducing.isSeparable_preimage hs | theorem | Topology.IsEmbedding.isSeparable_preimage | Topology.MetricSpace.Pseudo | Mathlib/Topology/MetricSpace/Pseudo/Basic.lean | [] | [
"IsSeparable",
"Set",
"TopologicalSpace"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
IsCompact.isSeparable {α : Type*} [TopologicalSpace α] [PseudoMetrizableSpace α]
{s : Set α} (hs : IsCompact s) : IsSeparable s | haveI : CompactSpace s := isCompact_iff_compactSpace.mp hs
.of_subtype s | theorem | IsCompact.isSeparable | Topology.MetricSpace.Pseudo | Mathlib/Topology/MetricSpace/Pseudo/Basic.lean | [] | [
"CompactSpace",
"IsCompact",
"IsSeparable",
"Set",
"TopologicalSpace"
] | A compact set is separable. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
secondCountable_of_almost_dense_set
(H : ∀ ε > (0 : ℝ), ∃ s : Set α, s.Countable ∧ ∀ x, ∃ y ∈ s, dist x y ≤ ε) :
SecondCountableTopology α | by
refine EMetric.secondCountable_of_almost_dense_set fun ε ε0 => ?_
rcases ENNReal.lt_iff_exists_nnreal_btwn.1 ε0 with ⟨ε', ε'0, ε'ε⟩
choose s hsc y hys hyx using H ε' (mod_cast ε'0)
refine ⟨s, hsc, iUnion₂_eq_univ_iff.2 fun x => ⟨y x, hys _, le_trans ?_ ε'ε.le⟩⟩
exact mod_cast hyx x | theorem | Metric.secondCountable_of_almost_dense_set | Topology.MetricSpace.Pseudo | Mathlib/Topology/MetricSpace/Pseudo/Basic.lean | [] | [
"EMetric.secondCountable_of_almost_dense_set",
"SecondCountableTopology",
"Set",
"le_trans"
] | A pseudometric space is second countable if, for every `ε > 0`, there is a countable set which
is `ε`-dense. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
finite_cover_balls_of_compact (hs : IsCompact s) {e : ℝ} (he : 0 < e) :
∃ t ⊆ s, t.Finite ∧ s ⊆ ⋃ x ∈ t, ball x e | let ⟨t, hts, ht⟩ := hs.elim_nhds_subcover _ (fun x _ => ball_mem_nhds x he)
⟨t, hts, t.finite_toSet, ht⟩
alias IsCompact.finite_cover_balls := finite_cover_balls_of_compact | theorem | finite_cover_balls_of_compact | Topology.MetricSpace.Pseudo | Mathlib/Topology/MetricSpace/Pseudo/Basic.lean | [] | [
"IsCompact"
] | Any compact set in a pseudometric space can be covered by finitely many balls of a given
positive radius | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
exists_finite_cover_balls_of_isCompact_closure (hs : IsCompact (closure s)) (hε : 0 < ε) :
∃ t ⊆ s, t.Finite ∧ s ⊆ ⋃ x ∈ t, ball x ε | by
obtain ⟨t, hst⟩ := hs.elim_finite_subcover (fun x : s ↦ ball x ε) (fun _ ↦ isOpen_ball) fun x hx ↦
let ⟨y, hy, hxy⟩ := Metric.mem_closure_iff.1 hx _ hε; mem_iUnion.2 ⟨⟨y, hy⟩, hxy⟩
refine ⟨t.map ⟨Subtype.val, Subtype.val_injective⟩, by simp, Finset.finite_toSet _, ?_⟩
simpa using subset_closure.trans hst | lemma | exists_finite_cover_balls_of_isCompact_closure | Topology.MetricSpace.Pseudo | Mathlib/Topology/MetricSpace/Pseudo/Basic.lean | [] | [
"Finset.finite_toSet",
"IsCompact",
"closure"
] | Any relatively compact set in a pseudometric space can be covered by finitely many balls of a
given positive radius. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
ContinuousOn.isSeparable_image {α : Type*} [TopologicalSpace α] [PseudoMetrizableSpace α]
[TopologicalSpace β] {f : α → β} {s : Set α}
(hf : ContinuousOn f s) (hs : IsSeparable s) : IsSeparable (f '' s) | by
rw [image_eq_range, ← image_univ]
exact (isSeparable_univ_iff.2 hs.separableSpace).image hf.restrict | theorem | ContinuousOn.isSeparable_image | Topology.MetricSpace.Pseudo | Mathlib/Topology/MetricSpace/Pseudo/Basic.lean | [] | [
"ContinuousOn",
"IsSeparable",
"Set",
"TopologicalSpace"
] | If a map is continuous on a separable set `s`, then the image of `s` is also separable. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
PseudoMetricSpace.induced {α β} (f : α → β) (m : PseudoMetricSpace β) :
PseudoMetricSpace α | where
dist x y := dist (f x) (f y)
dist_self _ := dist_self _
dist_comm _ _ := dist_comm _ _
dist_triangle _ _ _ := dist_triangle _ _ _
edist x y := edist (f x) (f y)
edist_dist _ _ := edist_dist _ _
toUniformSpace := UniformSpace.comap f m.toUniformSpace
uniformity_dist := (uniformity_basis_dist.comap ... | abbrev | PseudoMetricSpace.induced | Topology.MetricSpace.Pseudo | Mathlib/Topology/MetricSpace/Pseudo/Constructions.lean | [] | [
"Bornology.induced",
"PseudoMetricSpace",
"Set.ext",
"UniformSpace.comap",
"dist_comm",
"dist_self",
"dist_triangle",
"edist_dist"
] | Pseudometric space structure pulled back by a function. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
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