fact stringlengths 6 3.84k | type stringclasses 11
values | library stringclasses 32
values | imports listlengths 1 14 | filename stringlengths 20 95 | symbolic_name stringlengths 1 90 | docstring stringlengths 7 20k ⌀ |
|---|---|---|---|---|---|---|
@[nolint unusedArguments]
protected k (_hf : AntilipschitzWith K f) : ℝ≥0 := K | def | Topology | [
"Mathlib.Topology.UniformSpace.CompleteSeparated",
"Mathlib.Topology.EMetricSpace.Lipschitz",
"Mathlib.Topology.MetricSpace.Basic",
"Mathlib.Topology.MetricSpace.Bounded"
] | Mathlib/Topology/MetricSpace/Antilipschitz.lean | k | Extract the constant from `hf : AntilipschitzWith K f`. This is useful, e.g.,
if `K` is given by a long formula, and we want to reuse this value. |
protected injective {α : Type*} {β : Type*} [EMetricSpace α] [PseudoEMetricSpace β]
{K : ℝ≥0} {f : α → β} (hf : AntilipschitzWith K f) : Function.Injective f := fun x y h => by
simpa only [h, edist_self, mul_zero, edist_le_zero] using hf x y | theorem | Topology | [
"Mathlib.Topology.UniformSpace.CompleteSeparated",
"Mathlib.Topology.EMetricSpace.Lipschitz",
"Mathlib.Topology.MetricSpace.Basic",
"Mathlib.Topology.MetricSpace.Bounded"
] | Mathlib/Topology/MetricSpace/Antilipschitz.lean | injective | null |
mul_le_edist (hf : AntilipschitzWith K f) (x y : α) :
(K : ℝ≥0∞)⁻¹ * edist x y ≤ edist (f x) (f y) := by
rw [mul_comm, ← div_eq_mul_inv]
exact ENNReal.div_le_of_le_mul' (hf x y) | theorem | Topology | [
"Mathlib.Topology.UniformSpace.CompleteSeparated",
"Mathlib.Topology.EMetricSpace.Lipschitz",
"Mathlib.Topology.MetricSpace.Basic",
"Mathlib.Topology.MetricSpace.Bounded"
] | Mathlib/Topology/MetricSpace/Antilipschitz.lean | mul_le_edist | null |
ediam_preimage_le (hf : AntilipschitzWith K f) (s : Set β) : diam (f ⁻¹' s) ≤ K * diam s :=
diam_le fun x hx y hy => (hf x y).trans <|
mul_le_mul_left' (edist_le_diam_of_mem (mem_preimage.1 hx) hy) K | theorem | Topology | [
"Mathlib.Topology.UniformSpace.CompleteSeparated",
"Mathlib.Topology.EMetricSpace.Lipschitz",
"Mathlib.Topology.MetricSpace.Basic",
"Mathlib.Topology.MetricSpace.Bounded"
] | Mathlib/Topology/MetricSpace/Antilipschitz.lean | ediam_preimage_le | null |
le_mul_ediam_image (hf : AntilipschitzWith K f) (s : Set α) : diam s ≤ K * diam (f '' s) :=
(diam_mono (subset_preimage_image _ _)).trans (hf.ediam_preimage_le (f '' s)) | theorem | Topology | [
"Mathlib.Topology.UniformSpace.CompleteSeparated",
"Mathlib.Topology.EMetricSpace.Lipschitz",
"Mathlib.Topology.MetricSpace.Basic",
"Mathlib.Topology.MetricSpace.Bounded"
] | Mathlib/Topology/MetricSpace/Antilipschitz.lean | le_mul_ediam_image | null |
protected id : AntilipschitzWith 1 (id : α → α) := fun x y => by
simp only [ENNReal.coe_one, one_mul, id, le_refl] | theorem | Topology | [
"Mathlib.Topology.UniformSpace.CompleteSeparated",
"Mathlib.Topology.EMetricSpace.Lipschitz",
"Mathlib.Topology.MetricSpace.Basic",
"Mathlib.Topology.MetricSpace.Bounded"
] | Mathlib/Topology/MetricSpace/Antilipschitz.lean | id | null |
comp {Kg : ℝ≥0} {g : β → γ} (hg : AntilipschitzWith Kg g) {Kf : ℝ≥0} {f : α → β}
(hf : AntilipschitzWith Kf f) : AntilipschitzWith (Kf * Kg) (g ∘ f) := fun x y =>
calc
edist x y ≤ Kf * edist (f x) (f y) := hf x y
_ ≤ Kf * (Kg * edist (g (f x)) (g (f y))) := mul_right_mono (hg _ _)
_ = _ := by rw [ENNR... | theorem | Topology | [
"Mathlib.Topology.UniformSpace.CompleteSeparated",
"Mathlib.Topology.EMetricSpace.Lipschitz",
"Mathlib.Topology.MetricSpace.Basic",
"Mathlib.Topology.MetricSpace.Bounded"
] | Mathlib/Topology/MetricSpace/Antilipschitz.lean | comp | null |
restrict (hf : AntilipschitzWith K f) (s : Set α) : AntilipschitzWith K (s.restrict f) :=
fun x y => hf x y | theorem | Topology | [
"Mathlib.Topology.UniformSpace.CompleteSeparated",
"Mathlib.Topology.EMetricSpace.Lipschitz",
"Mathlib.Topology.MetricSpace.Basic",
"Mathlib.Topology.MetricSpace.Bounded"
] | Mathlib/Topology/MetricSpace/Antilipschitz.lean | restrict | null |
codRestrict (hf : AntilipschitzWith K f) {s : Set β} (hs : ∀ x, f x ∈ s) :
AntilipschitzWith K (s.codRestrict f hs) := fun x y => hf x y | theorem | Topology | [
"Mathlib.Topology.UniformSpace.CompleteSeparated",
"Mathlib.Topology.EMetricSpace.Lipschitz",
"Mathlib.Topology.MetricSpace.Basic",
"Mathlib.Topology.MetricSpace.Bounded"
] | Mathlib/Topology/MetricSpace/Antilipschitz.lean | codRestrict | null |
to_rightInvOn' {s : Set α} (hf : AntilipschitzWith K (s.restrict f)) {g : β → α}
{t : Set β} (g_maps : MapsTo g t s) (g_inv : RightInvOn g f t) :
LipschitzWith K (t.restrict g) := fun x y => by
simpa only [restrict_apply, g_inv x.mem, g_inv y.mem, Subtype.edist_mk_mk]
using hf ⟨g x, g_maps x.mem⟩ ⟨g y, g_... | theorem | Topology | [
"Mathlib.Topology.UniformSpace.CompleteSeparated",
"Mathlib.Topology.EMetricSpace.Lipschitz",
"Mathlib.Topology.MetricSpace.Basic",
"Mathlib.Topology.MetricSpace.Bounded"
] | Mathlib/Topology/MetricSpace/Antilipschitz.lean | to_rightInvOn' | null |
to_rightInvOn (hf : AntilipschitzWith K f) {g : β → α} {t : Set β} (h : RightInvOn g f t) :
LipschitzWith K (t.restrict g) :=
(hf.restrict univ).to_rightInvOn' (mapsTo_univ g t) h | theorem | Topology | [
"Mathlib.Topology.UniformSpace.CompleteSeparated",
"Mathlib.Topology.EMetricSpace.Lipschitz",
"Mathlib.Topology.MetricSpace.Basic",
"Mathlib.Topology.MetricSpace.Bounded"
] | Mathlib/Topology/MetricSpace/Antilipschitz.lean | to_rightInvOn | null |
to_rightInverse (hf : AntilipschitzWith K f) {g : β → α} (hg : Function.RightInverse g f) :
LipschitzWith K g := by
intro x y
have := hf (g x) (g y)
rwa [hg x, hg y] at this | theorem | Topology | [
"Mathlib.Topology.UniformSpace.CompleteSeparated",
"Mathlib.Topology.EMetricSpace.Lipschitz",
"Mathlib.Topology.MetricSpace.Basic",
"Mathlib.Topology.MetricSpace.Bounded"
] | Mathlib/Topology/MetricSpace/Antilipschitz.lean | to_rightInverse | null |
comap_uniformity_le (hf : AntilipschitzWith K f) : (𝓤 β).comap (Prod.map f f) ≤ 𝓤 α := by
refine ((uniformity_basis_edist.comap _).le_basis_iff uniformity_basis_edist).2 fun ε h₀ => ?_
refine ⟨(↑K)⁻¹ * ε, ENNReal.mul_pos (ENNReal.inv_ne_zero.2 ENNReal.coe_ne_top) h₀.ne', ?_⟩
refine fun x hx => (hf x.1 x.2).tran... | theorem | Topology | [
"Mathlib.Topology.UniformSpace.CompleteSeparated",
"Mathlib.Topology.EMetricSpace.Lipschitz",
"Mathlib.Topology.MetricSpace.Basic",
"Mathlib.Topology.MetricSpace.Bounded"
] | Mathlib/Topology/MetricSpace/Antilipschitz.lean | comap_uniformity_le | null |
isUniformInducing (hf : AntilipschitzWith K f) (hfc : UniformContinuous f) :
IsUniformInducing f :=
⟨le_antisymm hf.comap_uniformity_le hfc.le_comap⟩ | theorem | Topology | [
"Mathlib.Topology.UniformSpace.CompleteSeparated",
"Mathlib.Topology.EMetricSpace.Lipschitz",
"Mathlib.Topology.MetricSpace.Basic",
"Mathlib.Topology.MetricSpace.Bounded"
] | Mathlib/Topology/MetricSpace/Antilipschitz.lean | isUniformInducing | null |
isUniformEmbedding {α β : Type*} [EMetricSpace α] [PseudoEMetricSpace β] {K : ℝ≥0} {f : α → β}
(hf : AntilipschitzWith K f) (hfc : UniformContinuous f) : IsUniformEmbedding f :=
⟨hf.isUniformInducing hfc, hf.injective⟩ | lemma | Topology | [
"Mathlib.Topology.UniformSpace.CompleteSeparated",
"Mathlib.Topology.EMetricSpace.Lipschitz",
"Mathlib.Topology.MetricSpace.Basic",
"Mathlib.Topology.MetricSpace.Bounded"
] | Mathlib/Topology/MetricSpace/Antilipschitz.lean | isUniformEmbedding | null |
isComplete_range [CompleteSpace α] (hf : AntilipschitzWith K f)
(hfc : UniformContinuous f) : IsComplete (range f) :=
(hf.isUniformInducing hfc).isComplete_range | theorem | Topology | [
"Mathlib.Topology.UniformSpace.CompleteSeparated",
"Mathlib.Topology.EMetricSpace.Lipschitz",
"Mathlib.Topology.MetricSpace.Basic",
"Mathlib.Topology.MetricSpace.Bounded"
] | Mathlib/Topology/MetricSpace/Antilipschitz.lean | isComplete_range | null |
isClosed_range {α β : Type*} [PseudoEMetricSpace α] [EMetricSpace β] [CompleteSpace α]
{f : α → β} {K : ℝ≥0} (hf : AntilipschitzWith K f) (hfc : UniformContinuous f) :
IsClosed (range f) :=
(hf.isComplete_range hfc).isClosed | theorem | Topology | [
"Mathlib.Topology.UniformSpace.CompleteSeparated",
"Mathlib.Topology.EMetricSpace.Lipschitz",
"Mathlib.Topology.MetricSpace.Basic",
"Mathlib.Topology.MetricSpace.Bounded"
] | Mathlib/Topology/MetricSpace/Antilipschitz.lean | isClosed_range | null |
isClosedEmbedding {α : Type*} {β : Type*} [EMetricSpace α] [EMetricSpace β] {K : ℝ≥0}
{f : α → β} [CompleteSpace α] (hf : AntilipschitzWith K f) (hfc : UniformContinuous f) :
IsClosedEmbedding f :=
{ (hf.isUniformEmbedding hfc).isEmbedding with isClosed_range := hf.isClosed_range hfc } | theorem | Topology | [
"Mathlib.Topology.UniformSpace.CompleteSeparated",
"Mathlib.Topology.EMetricSpace.Lipschitz",
"Mathlib.Topology.MetricSpace.Basic",
"Mathlib.Topology.MetricSpace.Bounded"
] | Mathlib/Topology/MetricSpace/Antilipschitz.lean | isClosedEmbedding | null |
subtype_coe (s : Set α) : AntilipschitzWith 1 ((↑) : s → α) :=
AntilipschitzWith.id.restrict s
@[nontriviality] | theorem | Topology | [
"Mathlib.Topology.UniformSpace.CompleteSeparated",
"Mathlib.Topology.EMetricSpace.Lipschitz",
"Mathlib.Topology.MetricSpace.Basic",
"Mathlib.Topology.MetricSpace.Bounded"
] | Mathlib/Topology/MetricSpace/Antilipschitz.lean | subtype_coe | null |
of_subsingleton [Subsingleton α] {K : ℝ≥0} : AntilipschitzWith K f := fun x y => by
simp only [Subsingleton.elim x y, edist_self, zero_le] | theorem | Topology | [
"Mathlib.Topology.UniformSpace.CompleteSeparated",
"Mathlib.Topology.EMetricSpace.Lipschitz",
"Mathlib.Topology.MetricSpace.Basic",
"Mathlib.Topology.MetricSpace.Bounded"
] | Mathlib/Topology/MetricSpace/Antilipschitz.lean | of_subsingleton | null |
protected subsingleton {α β} [EMetricSpace α] [PseudoEMetricSpace β] {f : α → β}
(h : AntilipschitzWith 0 f) : Subsingleton α :=
⟨fun x y => edist_le_zero.1 <| (h x y).trans_eq <| zero_mul _⟩ | theorem | Topology | [
"Mathlib.Topology.UniformSpace.CompleteSeparated",
"Mathlib.Topology.EMetricSpace.Lipschitz",
"Mathlib.Topology.MetricSpace.Basic",
"Mathlib.Topology.MetricSpace.Bounded"
] | Mathlib/Topology/MetricSpace/Antilipschitz.lean | subsingleton | If `f : α → β` is `0`-antilipschitz, then `α` is a `subsingleton`. |
isBounded_preimage (hf : AntilipschitzWith K f) {s : Set β} (hs : IsBounded s) :
IsBounded (f ⁻¹' s) :=
isBounded_iff_ediam_ne_top.2 <| ne_top_of_le_ne_top
(ENNReal.mul_ne_top ENNReal.coe_ne_top hs.ediam_ne_top) (hf.ediam_preimage_le _) | theorem | Topology | [
"Mathlib.Topology.UniformSpace.CompleteSeparated",
"Mathlib.Topology.EMetricSpace.Lipschitz",
"Mathlib.Topology.MetricSpace.Basic",
"Mathlib.Topology.MetricSpace.Bounded"
] | Mathlib/Topology/MetricSpace/Antilipschitz.lean | isBounded_preimage | null |
tendsto_cobounded (hf : AntilipschitzWith K f) : Tendsto f (cobounded α) (cobounded β) :=
compl_surjective.forall.2 fun _ ↦ hf.isBounded_preimage | theorem | Topology | [
"Mathlib.Topology.UniformSpace.CompleteSeparated",
"Mathlib.Topology.EMetricSpace.Lipschitz",
"Mathlib.Topology.MetricSpace.Basic",
"Mathlib.Topology.MetricSpace.Bounded"
] | Mathlib/Topology/MetricSpace/Antilipschitz.lean | tendsto_cobounded | null |
protected properSpace {α : Type*} [MetricSpace α] {K : ℝ≥0} {f : α → β} [ProperSpace α]
(hK : AntilipschitzWith K f) (f_cont : Continuous f) (hf : Function.Surjective f) :
ProperSpace β := by
refine ⟨fun x₀ r => ?_⟩
let K := f ⁻¹' closedBall x₀ r
have A : IsClosed K := isClosed_closedBall.preimage f_cont
... | theorem | Topology | [
"Mathlib.Topology.UniformSpace.CompleteSeparated",
"Mathlib.Topology.EMetricSpace.Lipschitz",
"Mathlib.Topology.MetricSpace.Basic",
"Mathlib.Topology.MetricSpace.Bounded"
] | Mathlib/Topology/MetricSpace/Antilipschitz.lean | properSpace | The image of a proper space under an expanding onto map is proper. |
isBounded_of_image2_left (f : α → β → γ) {K₁ : ℝ≥0}
(hf : ∀ b, AntilipschitzWith K₁ fun a => f a b) {s : Set α} {t : Set β}
(hst : IsBounded (Set.image2 f s t)) : IsBounded s ∨ IsBounded t := by
contrapose! hst
obtain ⟨b, hb⟩ : t.Nonempty := nonempty_of_not_isBounded hst.2
have : ¬IsBounded (Set.image2 f ... | theorem | Topology | [
"Mathlib.Topology.UniformSpace.CompleteSeparated",
"Mathlib.Topology.EMetricSpace.Lipschitz",
"Mathlib.Topology.MetricSpace.Basic",
"Mathlib.Topology.MetricSpace.Bounded"
] | Mathlib/Topology/MetricSpace/Antilipschitz.lean | isBounded_of_image2_left | null |
isBounded_of_image2_right {f : α → β → γ} {K₂ : ℝ≥0} (hf : ∀ a, AntilipschitzWith K₂ (f a))
{s : Set α} {t : Set β} (hst : IsBounded (Set.image2 f s t)) : IsBounded s ∨ IsBounded t :=
Or.symm <| isBounded_of_image2_left (flip f) hf <| image2_swap f s t ▸ hst | theorem | Topology | [
"Mathlib.Topology.UniformSpace.CompleteSeparated",
"Mathlib.Topology.EMetricSpace.Lipschitz",
"Mathlib.Topology.MetricSpace.Basic",
"Mathlib.Topology.MetricSpace.Bounded"
] | Mathlib/Topology/MetricSpace/Antilipschitz.lean | isBounded_of_image2_right | null |
LipschitzWith.to_rightInverse [PseudoEMetricSpace α] [PseudoEMetricSpace β] {K : ℝ≥0}
{f : α → β} (hf : LipschitzWith K f) {g : β → α} (hg : Function.RightInverse g f) :
AntilipschitzWith K g := fun x y => by simpa only [hg _] using hf (g x) (g y) | theorem | Topology | [
"Mathlib.Topology.UniformSpace.CompleteSeparated",
"Mathlib.Topology.EMetricSpace.Lipschitz",
"Mathlib.Topology.MetricSpace.Basic",
"Mathlib.Topology.MetricSpace.Bounded"
] | Mathlib/Topology/MetricSpace/Antilipschitz.lean | LipschitzWith.to_rightInverse | null |
isUniformEmbedding_iff' [PseudoMetricSpace β] {f : γ → β} :
IsUniformEmbedding f ↔
(∀ ε > 0, ∃ δ > 0, ∀ {a b : γ}, dist a b < δ → dist (f a) (f b) < ε) ∧
∀ δ > 0, ∃ ε > 0, ∀ {a b : γ}, dist (f a) (f b) < ε → dist a b < δ := by
rw [isUniformEmbedding_iff_isUniformInducing, isUniformInducing_iff, unif... | theorem | Topology | [
"Mathlib.Topology.MetricSpace.Pseudo.Basic",
"Mathlib.Topology.MetricSpace.Pseudo.Lemmas",
"Mathlib.Topology.MetricSpace.Pseudo.Pi",
"Mathlib.Topology.MetricSpace.Defs"
] | Mathlib/Topology/MetricSpace/Basic.lean | isUniformEmbedding_iff' | A map between metric spaces is a uniform embedding if and only if the distance between `f x`
and `f y` is controlled in terms of the distance between `x` and `y` and conversely. |
_root_.MetricSpace.ofT0PseudoMetricSpace (α : Type*) [PseudoMetricSpace α] [T0Space α] :
MetricSpace α where
toPseudoMetricSpace := ‹_›
eq_of_dist_eq_zero hdist := (Metric.inseparable_iff.2 hdist).eq | abbrev | Topology | [
"Mathlib.Topology.MetricSpace.Pseudo.Basic",
"Mathlib.Topology.MetricSpace.Pseudo.Lemmas",
"Mathlib.Topology.MetricSpace.Pseudo.Pi",
"Mathlib.Topology.MetricSpace.Defs"
] | Mathlib/Topology/MetricSpace/Basic.lean | _root_.MetricSpace.ofT0PseudoMetricSpace | If a `PseudoMetricSpace` is a T₀ space, then it is a `MetricSpace`. |
isUniformEmbedding_bot_of_pairwise_le_dist {β : Type*} {ε : ℝ} (hε : 0 < ε) {f : β → α}
(hf : Pairwise fun x y => ε ≤ dist (f x) (f y)) :
@IsUniformEmbedding _ _ ⊥ (by infer_instance) f :=
isUniformEmbedding_of_spaced_out (dist_mem_uniformity hε) <| by simpa using hf | theorem | Topology | [
"Mathlib.Topology.MetricSpace.Pseudo.Basic",
"Mathlib.Topology.MetricSpace.Pseudo.Lemmas",
"Mathlib.Topology.MetricSpace.Pseudo.Pi",
"Mathlib.Topology.MetricSpace.Defs"
] | Mathlib/Topology/MetricSpace/Basic.lean | isUniformEmbedding_bot_of_pairwise_le_dist | A metric space induces an emetric space -/
instance (priority := 100) _root_.MetricSpace.toEMetricSpace : EMetricSpace γ :=
.ofT0PseudoEMetricSpace γ
theorem isClosed_of_pairwise_le_dist {s : Set γ} {ε : ℝ} (hε : 0 < ε)
(hs : s.Pairwise fun x y => ε ≤ dist x y) : IsClosed s :=
isClosed_of_spaced_out (dist_mem_... |
EMetricSpace.toMetricSpaceOfDist {α : Type u} [EMetricSpace α] (dist : α → α → ℝ)
(edist_ne_top : ∀ x y : α, edist x y ≠ ⊤) (h : ∀ x y, dist x y = ENNReal.toReal (edist x y)) :
MetricSpace α :=
@MetricSpace.ofT0PseudoMetricSpace _
(PseudoEMetricSpace.toPseudoMetricSpaceOfDist dist edist_ne_top h) _ | abbrev | Topology | [
"Mathlib.Topology.MetricSpace.Pseudo.Basic",
"Mathlib.Topology.MetricSpace.Pseudo.Lemmas",
"Mathlib.Topology.MetricSpace.Pseudo.Pi",
"Mathlib.Topology.MetricSpace.Defs"
] | Mathlib/Topology/MetricSpace/Basic.lean | EMetricSpace.toMetricSpaceOfDist | One gets a metric space from an emetric space if the edistance
is everywhere finite, by pushing the edistance to reals. We set it up so that the edist and the
uniformity are defeq in the metric space and the emetric space. In this definition, the distance
is given separately, to be able to prescribe some expression whi... |
EMetricSpace.toMetricSpace {α : Type u} [EMetricSpace α] (h : ∀ x y : α, edist x y ≠ ⊤) :
MetricSpace α :=
EMetricSpace.toMetricSpaceOfDist (fun x y => ENNReal.toReal (edist x y)) h fun _ _ => rfl | def | Topology | [
"Mathlib.Topology.MetricSpace.Pseudo.Basic",
"Mathlib.Topology.MetricSpace.Pseudo.Lemmas",
"Mathlib.Topology.MetricSpace.Pseudo.Pi",
"Mathlib.Topology.MetricSpace.Defs"
] | Mathlib/Topology/MetricSpace/Basic.lean | EMetricSpace.toMetricSpace | One gets a metric space from an emetric space if the edistance
is everywhere finite, by pushing the edistance to reals. We set it up so that the edist and the
uniformity are defeq in the metric space and the emetric space. |
MetricSpace.induced {γ β} (f : γ → β) (hf : Function.Injective f) (m : MetricSpace β) :
MetricSpace γ :=
{ PseudoMetricSpace.induced f m.toPseudoMetricSpace with
eq_of_dist_eq_zero := fun h => hf (dist_eq_zero.1 h) } | abbrev | Topology | [
"Mathlib.Topology.MetricSpace.Pseudo.Basic",
"Mathlib.Topology.MetricSpace.Pseudo.Lemmas",
"Mathlib.Topology.MetricSpace.Pseudo.Pi",
"Mathlib.Topology.MetricSpace.Defs"
] | Mathlib/Topology/MetricSpace/Basic.lean | MetricSpace.induced | Metric space structure pulled back by an injective function. Injectivity is necessary to
ensure that `dist x y = 0` only if `x = y`. |
IsUniformEmbedding.comapMetricSpace {α β} [UniformSpace α] [m : MetricSpace β] (f : α → β)
(h : IsUniformEmbedding f) : MetricSpace α :=
.replaceUniformity (.induced f h.injective m) h.comap_uniformity.symm | abbrev | Topology | [
"Mathlib.Topology.MetricSpace.Pseudo.Basic",
"Mathlib.Topology.MetricSpace.Pseudo.Lemmas",
"Mathlib.Topology.MetricSpace.Pseudo.Pi",
"Mathlib.Topology.MetricSpace.Defs"
] | Mathlib/Topology/MetricSpace/Basic.lean | IsUniformEmbedding.comapMetricSpace | Pull back a metric space structure by a uniform embedding. This is a version of
`MetricSpace.induced` useful in case if the domain already has a `UniformSpace` structure. |
Topology.IsEmbedding.comapMetricSpace {α β} [TopologicalSpace α] [m : MetricSpace β]
(f : α → β) (h : IsEmbedding f) : MetricSpace α :=
.replaceTopology (.induced f h.injective m) h.eq_induced | abbrev | Topology | [
"Mathlib.Topology.MetricSpace.Pseudo.Basic",
"Mathlib.Topology.MetricSpace.Pseudo.Lemmas",
"Mathlib.Topology.MetricSpace.Pseudo.Pi",
"Mathlib.Topology.MetricSpace.Defs"
] | Mathlib/Topology/MetricSpace/Basic.lean | Topology.IsEmbedding.comapMetricSpace | Pull back a metric space structure by an embedding. This is a version of
`MetricSpace.induced` useful in case if the domain already has a `TopologicalSpace` structure. |
Subtype.metricSpace {α : Type*} {p : α → Prop} [MetricSpace α] :
MetricSpace (Subtype p) :=
.induced Subtype.val Subtype.coe_injective ‹_›
@[to_additive] | instance | Topology | [
"Mathlib.Topology.MetricSpace.Pseudo.Basic",
"Mathlib.Topology.MetricSpace.Pseudo.Lemmas",
"Mathlib.Topology.MetricSpace.Pseudo.Pi",
"Mathlib.Topology.MetricSpace.Defs"
] | Mathlib/Topology/MetricSpace/Basic.lean | Subtype.metricSpace | null |
MulOpposite.instMetricSpace {α : Type*} [MetricSpace α] : MetricSpace αᵐᵒᵖ :=
MetricSpace.induced MulOpposite.unop MulOpposite.unop_injective ‹_› | instance | Topology | [
"Mathlib.Topology.MetricSpace.Pseudo.Basic",
"Mathlib.Topology.MetricSpace.Pseudo.Lemmas",
"Mathlib.Topology.MetricSpace.Pseudo.Pi",
"Mathlib.Topology.MetricSpace.Defs"
] | Mathlib/Topology/MetricSpace/Basic.lean | MulOpposite.instMetricSpace | null |
Real.metricSpace : MetricSpace ℝ := .ofT0PseudoMetricSpace ℝ | instance | Topology | [
"Mathlib.Topology.MetricSpace.Pseudo.Basic",
"Mathlib.Topology.MetricSpace.Pseudo.Lemmas",
"Mathlib.Topology.MetricSpace.Pseudo.Pi",
"Mathlib.Topology.MetricSpace.Defs"
] | Mathlib/Topology/MetricSpace/Basic.lean | Real.metricSpace | Instantiate the reals as a metric space. |
Prod.metricSpaceMax [MetricSpace β] : MetricSpace (γ × β) :=
.ofT0PseudoMetricSpace _ | instance | Topology | [
"Mathlib.Topology.MetricSpace.Pseudo.Basic",
"Mathlib.Topology.MetricSpace.Pseudo.Lemmas",
"Mathlib.Topology.MetricSpace.Pseudo.Pi",
"Mathlib.Topology.MetricSpace.Defs"
] | Mathlib/Topology/MetricSpace/Basic.lean | Prod.metricSpaceMax | null |
metricSpacePi : MetricSpace (∀ b, X b) := .ofT0PseudoMetricSpace _ | instance | Topology | [
"Mathlib.Topology.MetricSpace.Pseudo.Basic",
"Mathlib.Topology.MetricSpace.Pseudo.Lemmas",
"Mathlib.Topology.MetricSpace.Pseudo.Pi",
"Mathlib.Topology.MetricSpace.Defs"
] | Mathlib/Topology/MetricSpace/Basic.lean | metricSpacePi | A finite product of metric spaces is a metric space, with the sup distance. |
secondCountable_of_countable_discretization {α : Type u} [PseudoMetricSpace α]
(H : ∀ ε > (0 : ℝ), ∃ (β : Type*) (_ : Encodable β) (F : α → β),
∀ x y, F x = F y → dist x y ≤ ε) :
SecondCountableTopology α := by
refine secondCountable_of_almost_dense_set fun ε ε0 => ?_
rcases H ε ε0 with ⟨β, fβ, F, hF⟩... | theorem | Topology | [
"Mathlib.Topology.MetricSpace.Pseudo.Basic",
"Mathlib.Topology.MetricSpace.Pseudo.Lemmas",
"Mathlib.Topology.MetricSpace.Pseudo.Pi",
"Mathlib.Topology.MetricSpace.Defs"
] | Mathlib/Topology/MetricSpace/Basic.lean | secondCountable_of_countable_discretization | A metric space is second countable if one can reconstruct up to any `ε>0` any element of the
space from countably many data. |
SeparationQuotient.instDist {α : Type u} [PseudoMetricSpace α] :
Dist (SeparationQuotient α) where
dist := lift₂ dist fun x y x' y' hx hy ↦ by rw [dist_edist, dist_edist, ← edist_mk x,
← edist_mk x', mk_eq_mk.2 hx, mk_eq_mk.2 hy] | instance | Topology | [
"Mathlib.Topology.MetricSpace.Pseudo.Basic",
"Mathlib.Topology.MetricSpace.Pseudo.Lemmas",
"Mathlib.Topology.MetricSpace.Pseudo.Pi",
"Mathlib.Topology.MetricSpace.Defs"
] | Mathlib/Topology/MetricSpace/Basic.lean | SeparationQuotient.instDist | null |
SeparationQuotient.dist_mk {α : Type u} [PseudoMetricSpace α] (p q : α) :
dist (mk p) (mk q) = dist p q :=
rfl | theorem | Topology | [
"Mathlib.Topology.MetricSpace.Pseudo.Basic",
"Mathlib.Topology.MetricSpace.Pseudo.Lemmas",
"Mathlib.Topology.MetricSpace.Pseudo.Pi",
"Mathlib.Topology.MetricSpace.Defs"
] | Mathlib/Topology/MetricSpace/Basic.lean | SeparationQuotient.dist_mk | null |
SeparationQuotient.instMetricSpace {α : Type u} [PseudoMetricSpace α] :
MetricSpace (SeparationQuotient α) :=
EMetricSpace.toMetricSpaceOfDist dist (surjective_mk.forall₂.2 edist_ne_top) <|
surjective_mk.forall₂.2 dist_edist | instance | Topology | [
"Mathlib.Topology.MetricSpace.Pseudo.Basic",
"Mathlib.Topology.MetricSpace.Pseudo.Lemmas",
"Mathlib.Topology.MetricSpace.Pseudo.Pi",
"Mathlib.Topology.MetricSpace.Defs"
] | Mathlib/Topology/MetricSpace/Basic.lean | SeparationQuotient.instMetricSpace | null |
uniformity_eq_of_bilipschitz (hf₁ : AntilipschitzWith K₁ f) (hf₂ : LipschitzWith K₂ f) :
𝓤[(inferInstance : UniformSpace β).comap f] = 𝓤 α :=
hf₁.isUniformInducing hf₂.uniformContinuous |>.comap_uniformity | lemma | Topology | [
"Mathlib.Topology.MetricSpace.Antilipschitz",
"Mathlib.Topology.MetricSpace.Lipschitz"
] | Mathlib/Topology/MetricSpace/Bilipschitz.lean | uniformity_eq_of_bilipschitz | If `f : α → β` is bilipschitz, then the pullback of the uniformity on `β` through `f` agrees
with the uniformity on `α`.
This can be used to provide the replacement equality when applying
`PseudoMetricSpace.replaceUniformity`, which can be useful when following the forgetful inheritance
pattern when creating type syno... |
bornology_eq_of_bilipschitz (hf₁ : AntilipschitzWith K₁ f) (hf₂ : LipschitzWith K₂ f) :
@cobounded _ (induced f) = cobounded α :=
le_antisymm hf₂.comap_cobounded_le hf₁.tendsto_cobounded.le_comap | lemma | Topology | [
"Mathlib.Topology.MetricSpace.Antilipschitz",
"Mathlib.Topology.MetricSpace.Lipschitz"
] | Mathlib/Topology/MetricSpace/Bilipschitz.lean | bornology_eq_of_bilipschitz | If `f : α → β` is bilipschitz, then the pullback of the bornology on `β` through `f` agrees
with the bornology on `α`. |
isBounded_iff_of_bilipschitz (hf₁ : AntilipschitzWith K₁ f) (hf₂ : LipschitzWith K₂ f)
(s : Set α) : @IsBounded _ (induced f) s ↔ Bornology.IsBounded s :=
Filter.ext_iff.1 (bornology_eq_of_bilipschitz hf₁ hf₂) (sᶜ) | lemma | Topology | [
"Mathlib.Topology.MetricSpace.Antilipschitz",
"Mathlib.Topology.MetricSpace.Lipschitz"
] | Mathlib/Topology/MetricSpace/Bilipschitz.lean | isBounded_iff_of_bilipschitz | If `f : α → β` is bilipschitz, then the pullback of the bornology on `β` through `f` agrees
with the bornology on `α`.
This can be used to provide the replacement equality when applying
`PseudoMetricSpace.replaceBornology`, which can be useful when following the forgetful inheritance
pattern when creating type synonym... |
totallyBounded_Icc (a b : α) : TotallyBounded (Icc a b) :=
isCompact_Icc.totallyBounded | lemma | Topology | [
"Mathlib.Topology.Order.Bornology",
"Mathlib.Topology.Order.Compact",
"Mathlib.Topology.MetricSpace.ProperSpace",
"Mathlib.Topology.MetricSpace.Cauchy",
"Mathlib.Topology.MetricSpace.Defs",
"Mathlib.Topology.EMetricSpace.Diam"
] | Mathlib/Topology/MetricSpace/Bounded.lean | totallyBounded_Icc | null |
totallyBounded_Ico (a b : α) : TotallyBounded (Ico a b) :=
(totallyBounded_Icc a b).subset Ico_subset_Icc_self | lemma | Topology | [
"Mathlib.Topology.Order.Bornology",
"Mathlib.Topology.Order.Compact",
"Mathlib.Topology.MetricSpace.ProperSpace",
"Mathlib.Topology.MetricSpace.Cauchy",
"Mathlib.Topology.MetricSpace.Defs",
"Mathlib.Topology.EMetricSpace.Diam"
] | Mathlib/Topology/MetricSpace/Bounded.lean | totallyBounded_Ico | null |
totallyBounded_Ioc (a b : α) : TotallyBounded (Ioc a b) :=
(totallyBounded_Icc a b).subset Ioc_subset_Icc_self | lemma | Topology | [
"Mathlib.Topology.Order.Bornology",
"Mathlib.Topology.Order.Compact",
"Mathlib.Topology.MetricSpace.ProperSpace",
"Mathlib.Topology.MetricSpace.Cauchy",
"Mathlib.Topology.MetricSpace.Defs",
"Mathlib.Topology.EMetricSpace.Diam"
] | Mathlib/Topology/MetricSpace/Bounded.lean | totallyBounded_Ioc | null |
totallyBounded_Ioo (a b : α) : TotallyBounded (Ioo a b) :=
(totallyBounded_Icc a b).subset Ioo_subset_Icc_self | lemma | Topology | [
"Mathlib.Topology.Order.Bornology",
"Mathlib.Topology.Order.Compact",
"Mathlib.Topology.MetricSpace.ProperSpace",
"Mathlib.Topology.MetricSpace.Cauchy",
"Mathlib.Topology.MetricSpace.Defs",
"Mathlib.Topology.EMetricSpace.Diam"
] | Mathlib/Topology/MetricSpace/Bounded.lean | totallyBounded_Ioo | null |
isBounded_closedBall : IsBounded (closedBall x r) :=
isBounded_iff.2 ⟨r + r, fun y hy z hz =>
calc dist y z ≤ dist y x + dist z x := dist_triangle_right _ _ _
_ ≤ r + r := add_le_add hy hz⟩ | theorem | Topology | [
"Mathlib.Topology.Order.Bornology",
"Mathlib.Topology.Order.Compact",
"Mathlib.Topology.MetricSpace.ProperSpace",
"Mathlib.Topology.MetricSpace.Cauchy",
"Mathlib.Topology.MetricSpace.Defs",
"Mathlib.Topology.EMetricSpace.Diam"
] | Mathlib/Topology/MetricSpace/Bounded.lean | isBounded_closedBall | Closed balls are bounded |
isBounded_ball : IsBounded (ball x r) :=
isBounded_closedBall.subset ball_subset_closedBall | theorem | Topology | [
"Mathlib.Topology.Order.Bornology",
"Mathlib.Topology.Order.Compact",
"Mathlib.Topology.MetricSpace.ProperSpace",
"Mathlib.Topology.MetricSpace.Cauchy",
"Mathlib.Topology.MetricSpace.Defs",
"Mathlib.Topology.EMetricSpace.Diam"
] | Mathlib/Topology/MetricSpace/Bounded.lean | isBounded_ball | Open balls are bounded |
isBounded_sphere : IsBounded (sphere x r) :=
isBounded_closedBall.subset sphere_subset_closedBall | theorem | Topology | [
"Mathlib.Topology.Order.Bornology",
"Mathlib.Topology.Order.Compact",
"Mathlib.Topology.MetricSpace.ProperSpace",
"Mathlib.Topology.MetricSpace.Cauchy",
"Mathlib.Topology.MetricSpace.Defs",
"Mathlib.Topology.EMetricSpace.Diam"
] | Mathlib/Topology/MetricSpace/Bounded.lean | isBounded_sphere | Spheres are bounded |
isBounded_iff_subset_closedBall (c : α) : IsBounded s ↔ ∃ r, s ⊆ closedBall c r :=
⟨fun h ↦ (isBounded_iff.1 (h.insert c)).imp fun _r hr _x hx ↦ hr (.inr hx) (mem_insert _ _),
fun ⟨_r, hr⟩ ↦ isBounded_closedBall.subset hr⟩ | theorem | Topology | [
"Mathlib.Topology.Order.Bornology",
"Mathlib.Topology.Order.Compact",
"Mathlib.Topology.MetricSpace.ProperSpace",
"Mathlib.Topology.MetricSpace.Cauchy",
"Mathlib.Topology.MetricSpace.Defs",
"Mathlib.Topology.EMetricSpace.Diam"
] | Mathlib/Topology/MetricSpace/Bounded.lean | isBounded_iff_subset_closedBall | Given a point, a bounded subset is included in some ball around this point |
_root_.Bornology.IsBounded.subset_closedBall (h : IsBounded s) (c : α) :
∃ r, s ⊆ closedBall c r :=
(isBounded_iff_subset_closedBall c).1 h | theorem | Topology | [
"Mathlib.Topology.Order.Bornology",
"Mathlib.Topology.Order.Compact",
"Mathlib.Topology.MetricSpace.ProperSpace",
"Mathlib.Topology.MetricSpace.Cauchy",
"Mathlib.Topology.MetricSpace.Defs",
"Mathlib.Topology.EMetricSpace.Diam"
] | Mathlib/Topology/MetricSpace/Bounded.lean | _root_.Bornology.IsBounded.subset_closedBall | null |
_root_.Bornology.IsBounded.subset_ball_lt (h : IsBounded s) (a : ℝ) (c : α) :
∃ r, a < r ∧ s ⊆ ball c r :=
let ⟨r, hr⟩ := h.subset_closedBall c
⟨max r a + 1, (le_max_right _ _).trans_lt (lt_add_one _), hr.trans <| closedBall_subset_ball <|
(le_max_left _ _).trans_lt (lt_add_one _)⟩ | theorem | Topology | [
"Mathlib.Topology.Order.Bornology",
"Mathlib.Topology.Order.Compact",
"Mathlib.Topology.MetricSpace.ProperSpace",
"Mathlib.Topology.MetricSpace.Cauchy",
"Mathlib.Topology.MetricSpace.Defs",
"Mathlib.Topology.EMetricSpace.Diam"
] | Mathlib/Topology/MetricSpace/Bounded.lean | _root_.Bornology.IsBounded.subset_ball_lt | null |
_root_.Bornology.IsBounded.subset_ball (h : IsBounded s) (c : α) : ∃ r, s ⊆ ball c r :=
(h.subset_ball_lt 0 c).imp fun _ ↦ And.right | theorem | Topology | [
"Mathlib.Topology.Order.Bornology",
"Mathlib.Topology.Order.Compact",
"Mathlib.Topology.MetricSpace.ProperSpace",
"Mathlib.Topology.MetricSpace.Cauchy",
"Mathlib.Topology.MetricSpace.Defs",
"Mathlib.Topology.EMetricSpace.Diam"
] | Mathlib/Topology/MetricSpace/Bounded.lean | _root_.Bornology.IsBounded.subset_ball | null |
isBounded_iff_subset_ball (c : α) : IsBounded s ↔ ∃ r, s ⊆ ball c r :=
⟨(IsBounded.subset_ball · c), fun ⟨_r, hr⟩ ↦ isBounded_ball.subset hr⟩ | theorem | Topology | [
"Mathlib.Topology.Order.Bornology",
"Mathlib.Topology.Order.Compact",
"Mathlib.Topology.MetricSpace.ProperSpace",
"Mathlib.Topology.MetricSpace.Cauchy",
"Mathlib.Topology.MetricSpace.Defs",
"Mathlib.Topology.EMetricSpace.Diam"
] | Mathlib/Topology/MetricSpace/Bounded.lean | isBounded_iff_subset_ball | null |
_root_.Bornology.IsBounded.subset_closedBall_lt (h : IsBounded s) (a : ℝ) (c : α) :
∃ r, a < r ∧ s ⊆ closedBall c r :=
let ⟨r, har, hr⟩ := h.subset_ball_lt a c
⟨r, har, hr.trans ball_subset_closedBall⟩ | theorem | Topology | [
"Mathlib.Topology.Order.Bornology",
"Mathlib.Topology.Order.Compact",
"Mathlib.Topology.MetricSpace.ProperSpace",
"Mathlib.Topology.MetricSpace.Cauchy",
"Mathlib.Topology.MetricSpace.Defs",
"Mathlib.Topology.EMetricSpace.Diam"
] | Mathlib/Topology/MetricSpace/Bounded.lean | _root_.Bornology.IsBounded.subset_closedBall_lt | null |
isBounded_closure_of_isBounded (h : IsBounded s) : IsBounded (closure s) :=
let ⟨C, h⟩ := isBounded_iff.1 h
isBounded_iff.2 ⟨C, fun _a ha _b hb => isClosed_Iic.closure_subset <|
map_mem_closure₂ continuous_dist ha hb h⟩ | theorem | Topology | [
"Mathlib.Topology.Order.Bornology",
"Mathlib.Topology.Order.Compact",
"Mathlib.Topology.MetricSpace.ProperSpace",
"Mathlib.Topology.MetricSpace.Cauchy",
"Mathlib.Topology.MetricSpace.Defs",
"Mathlib.Topology.EMetricSpace.Diam"
] | Mathlib/Topology/MetricSpace/Bounded.lean | isBounded_closure_of_isBounded | null |
protected _root_.Bornology.IsBounded.closure (h : IsBounded s) : IsBounded (closure s) :=
isBounded_closure_of_isBounded h
@[simp] | theorem | Topology | [
"Mathlib.Topology.Order.Bornology",
"Mathlib.Topology.Order.Compact",
"Mathlib.Topology.MetricSpace.ProperSpace",
"Mathlib.Topology.MetricSpace.Cauchy",
"Mathlib.Topology.MetricSpace.Defs",
"Mathlib.Topology.EMetricSpace.Diam"
] | Mathlib/Topology/MetricSpace/Bounded.lean | _root_.Bornology.IsBounded.closure | null |
isBounded_closure_iff : IsBounded (closure s) ↔ IsBounded s :=
⟨fun h => h.subset subset_closure, fun h => h.closure⟩ | theorem | Topology | [
"Mathlib.Topology.Order.Bornology",
"Mathlib.Topology.Order.Compact",
"Mathlib.Topology.MetricSpace.ProperSpace",
"Mathlib.Topology.MetricSpace.Cauchy",
"Mathlib.Topology.MetricSpace.Defs",
"Mathlib.Topology.EMetricSpace.Diam"
] | Mathlib/Topology/MetricSpace/Bounded.lean | isBounded_closure_iff | null |
hasBasis_cobounded_compl_closedBall (c : α) :
(cobounded α).HasBasis (fun _ ↦ True) (fun r ↦ (closedBall c r)ᶜ) :=
⟨compl_surjective.forall.2 fun _ ↦ (isBounded_iff_subset_closedBall c).trans <| by simp⟩ | theorem | Topology | [
"Mathlib.Topology.Order.Bornology",
"Mathlib.Topology.Order.Compact",
"Mathlib.Topology.MetricSpace.ProperSpace",
"Mathlib.Topology.MetricSpace.Cauchy",
"Mathlib.Topology.MetricSpace.Defs",
"Mathlib.Topology.EMetricSpace.Diam"
] | Mathlib/Topology/MetricSpace/Bounded.lean | hasBasis_cobounded_compl_closedBall | null |
hasAntitoneBasis_cobounded_compl_closedBall (c : α) :
(cobounded α).HasAntitoneBasis (fun r ↦ (closedBall c r)ᶜ) :=
⟨Metric.hasBasis_cobounded_compl_closedBall _, fun _ _ hr _ ↦ by simpa using hr.trans_lt⟩ | theorem | Topology | [
"Mathlib.Topology.Order.Bornology",
"Mathlib.Topology.Order.Compact",
"Mathlib.Topology.MetricSpace.ProperSpace",
"Mathlib.Topology.MetricSpace.Cauchy",
"Mathlib.Topology.MetricSpace.Defs",
"Mathlib.Topology.EMetricSpace.Diam"
] | Mathlib/Topology/MetricSpace/Bounded.lean | hasAntitoneBasis_cobounded_compl_closedBall | null |
hasBasis_cobounded_compl_ball (c : α) :
(cobounded α).HasBasis (fun _ ↦ True) (fun r ↦ (ball c r)ᶜ) :=
⟨compl_surjective.forall.2 fun _ ↦ (isBounded_iff_subset_ball c).trans <| by simp⟩ | theorem | Topology | [
"Mathlib.Topology.Order.Bornology",
"Mathlib.Topology.Order.Compact",
"Mathlib.Topology.MetricSpace.ProperSpace",
"Mathlib.Topology.MetricSpace.Cauchy",
"Mathlib.Topology.MetricSpace.Defs",
"Mathlib.Topology.EMetricSpace.Diam"
] | Mathlib/Topology/MetricSpace/Bounded.lean | hasBasis_cobounded_compl_ball | null |
hasAntitoneBasis_cobounded_compl_ball (c : α) :
(cobounded α).HasAntitoneBasis (fun r ↦ (ball c r)ᶜ) :=
⟨Metric.hasBasis_cobounded_compl_ball _, fun _ _ hr _ ↦ by simpa using hr.trans⟩
@[simp] | theorem | Topology | [
"Mathlib.Topology.Order.Bornology",
"Mathlib.Topology.Order.Compact",
"Mathlib.Topology.MetricSpace.ProperSpace",
"Mathlib.Topology.MetricSpace.Cauchy",
"Mathlib.Topology.MetricSpace.Defs",
"Mathlib.Topology.EMetricSpace.Diam"
] | Mathlib/Topology/MetricSpace/Bounded.lean | hasAntitoneBasis_cobounded_compl_ball | null |
comap_dist_right_atTop (c : α) : comap (dist · c) atTop = cobounded α :=
(atTop_basis.comap _).eq_of_same_basis <| by
simpa only [compl_def, mem_ball, not_lt] using hasBasis_cobounded_compl_ball c
@[simp] | theorem | Topology | [
"Mathlib.Topology.Order.Bornology",
"Mathlib.Topology.Order.Compact",
"Mathlib.Topology.MetricSpace.ProperSpace",
"Mathlib.Topology.MetricSpace.Cauchy",
"Mathlib.Topology.MetricSpace.Defs",
"Mathlib.Topology.EMetricSpace.Diam"
] | Mathlib/Topology/MetricSpace/Bounded.lean | comap_dist_right_atTop | null |
comap_dist_left_atTop (c : α) : comap (dist c) atTop = cobounded α := by
simpa only [dist_comm _ c] using comap_dist_right_atTop c
@[simp] | theorem | Topology | [
"Mathlib.Topology.Order.Bornology",
"Mathlib.Topology.Order.Compact",
"Mathlib.Topology.MetricSpace.ProperSpace",
"Mathlib.Topology.MetricSpace.Cauchy",
"Mathlib.Topology.MetricSpace.Defs",
"Mathlib.Topology.EMetricSpace.Diam"
] | Mathlib/Topology/MetricSpace/Bounded.lean | comap_dist_left_atTop | null |
tendsto_dist_right_atTop_iff (c : α) {f : β → α} {l : Filter β} :
Tendsto (fun x ↦ dist (f x) c) l atTop ↔ Tendsto f l (cobounded α) := by
rw [← comap_dist_right_atTop c, tendsto_comap_iff, Function.comp_def]
@[simp] | theorem | Topology | [
"Mathlib.Topology.Order.Bornology",
"Mathlib.Topology.Order.Compact",
"Mathlib.Topology.MetricSpace.ProperSpace",
"Mathlib.Topology.MetricSpace.Cauchy",
"Mathlib.Topology.MetricSpace.Defs",
"Mathlib.Topology.EMetricSpace.Diam"
] | Mathlib/Topology/MetricSpace/Bounded.lean | tendsto_dist_right_atTop_iff | null |
tendsto_dist_left_atTop_iff (c : α) {f : β → α} {l : Filter β} :
Tendsto (fun x ↦ dist c (f x)) l atTop ↔ Tendsto f l (cobounded α) := by
simp only [dist_comm c, tendsto_dist_right_atTop_iff] | theorem | Topology | [
"Mathlib.Topology.Order.Bornology",
"Mathlib.Topology.Order.Compact",
"Mathlib.Topology.MetricSpace.ProperSpace",
"Mathlib.Topology.MetricSpace.Cauchy",
"Mathlib.Topology.MetricSpace.Defs",
"Mathlib.Topology.EMetricSpace.Diam"
] | Mathlib/Topology/MetricSpace/Bounded.lean | tendsto_dist_left_atTop_iff | null |
tendsto_dist_right_cobounded_atTop (c : α) : Tendsto (dist · c) (cobounded α) atTop :=
tendsto_iff_comap.2 (comap_dist_right_atTop c).ge | theorem | Topology | [
"Mathlib.Topology.Order.Bornology",
"Mathlib.Topology.Order.Compact",
"Mathlib.Topology.MetricSpace.ProperSpace",
"Mathlib.Topology.MetricSpace.Cauchy",
"Mathlib.Topology.MetricSpace.Defs",
"Mathlib.Topology.EMetricSpace.Diam"
] | Mathlib/Topology/MetricSpace/Bounded.lean | tendsto_dist_right_cobounded_atTop | null |
tendsto_dist_left_cobounded_atTop (c : α) : Tendsto (dist c) (cobounded α) atTop :=
tendsto_iff_comap.2 (comap_dist_left_atTop c).ge | theorem | Topology | [
"Mathlib.Topology.Order.Bornology",
"Mathlib.Topology.Order.Compact",
"Mathlib.Topology.MetricSpace.ProperSpace",
"Mathlib.Topology.MetricSpace.Cauchy",
"Mathlib.Topology.MetricSpace.Defs",
"Mathlib.Topology.EMetricSpace.Diam"
] | Mathlib/Topology/MetricSpace/Bounded.lean | tendsto_dist_left_cobounded_atTop | null |
_root_.TotallyBounded.isBounded {s : Set α} (h : TotallyBounded s) : IsBounded s :=
let ⟨_t, fint, subs⟩ := (totallyBounded_iff.mp h) 1 zero_lt_one
((isBounded_biUnion fint).2 fun _ _ => isBounded_ball).subset subs | theorem | Topology | [
"Mathlib.Topology.Order.Bornology",
"Mathlib.Topology.Order.Compact",
"Mathlib.Topology.MetricSpace.ProperSpace",
"Mathlib.Topology.MetricSpace.Cauchy",
"Mathlib.Topology.MetricSpace.Defs",
"Mathlib.Topology.EMetricSpace.Diam"
] | Mathlib/Topology/MetricSpace/Bounded.lean | _root_.TotallyBounded.isBounded | A totally bounded set is bounded |
@[aesop 50% apply, grind ←]
_root_.IsCompact.isBounded {s : Set α} (h : IsCompact s) : IsBounded s :=
h.totallyBounded.isBounded | theorem | Topology | [
"Mathlib.Topology.Order.Bornology",
"Mathlib.Topology.Order.Compact",
"Mathlib.Topology.MetricSpace.ProperSpace",
"Mathlib.Topology.MetricSpace.Cauchy",
"Mathlib.Topology.MetricSpace.Defs",
"Mathlib.Topology.EMetricSpace.Diam"
] | Mathlib/Topology/MetricSpace/Bounded.lean | _root_.IsCompact.isBounded | A compact set is bounded |
cobounded_le_cocompact : cobounded α ≤ cocompact α :=
hasBasis_cocompact.ge_iff.2 fun _s hs ↦ hs.isBounded | theorem | Topology | [
"Mathlib.Topology.Order.Bornology",
"Mathlib.Topology.Order.Compact",
"Mathlib.Topology.MetricSpace.ProperSpace",
"Mathlib.Topology.MetricSpace.Cauchy",
"Mathlib.Topology.MetricSpace.Defs",
"Mathlib.Topology.EMetricSpace.Diam"
] | Mathlib/Topology/MetricSpace/Bounded.lean | cobounded_le_cocompact | null |
isCobounded_iff_closedBall_compl_subset {s : Set α} (c : α) :
IsCobounded s ↔ ∃ (r : ℝ), (Metric.closedBall c r)ᶜ ⊆ s := by
rw [← isBounded_compl_iff, isBounded_iff_subset_closedBall c]
apply exists_congr
intro r
rw [compl_subset_comm] | theorem | Topology | [
"Mathlib.Topology.Order.Bornology",
"Mathlib.Topology.Order.Compact",
"Mathlib.Topology.MetricSpace.ProperSpace",
"Mathlib.Topology.MetricSpace.Cauchy",
"Mathlib.Topology.MetricSpace.Defs",
"Mathlib.Topology.EMetricSpace.Diam"
] | Mathlib/Topology/MetricSpace/Bounded.lean | isCobounded_iff_closedBall_compl_subset | null |
_root_.Bornology.IsCobounded.closedBall_compl_subset {s : Set α} (hs : IsCobounded s)
(c : α) : ∃ (r : ℝ), (Metric.closedBall c r)ᶜ ⊆ s :=
(isCobounded_iff_closedBall_compl_subset c).mp hs | theorem | Topology | [
"Mathlib.Topology.Order.Bornology",
"Mathlib.Topology.Order.Compact",
"Mathlib.Topology.MetricSpace.ProperSpace",
"Mathlib.Topology.MetricSpace.Cauchy",
"Mathlib.Topology.MetricSpace.Defs",
"Mathlib.Topology.EMetricSpace.Diam"
] | Mathlib/Topology/MetricSpace/Bounded.lean | _root_.Bornology.IsCobounded.closedBall_compl_subset | null |
closedBall_compl_subset_of_mem_cocompact {s : Set α} (hs : s ∈ cocompact α) (c : α) :
∃ (r : ℝ), (Metric.closedBall c r)ᶜ ⊆ s :=
IsCobounded.closedBall_compl_subset (cobounded_le_cocompact hs) c | theorem | Topology | [
"Mathlib.Topology.Order.Bornology",
"Mathlib.Topology.Order.Compact",
"Mathlib.Topology.MetricSpace.ProperSpace",
"Mathlib.Topology.MetricSpace.Cauchy",
"Mathlib.Topology.MetricSpace.Defs",
"Mathlib.Topology.EMetricSpace.Diam"
] | Mathlib/Topology/MetricSpace/Bounded.lean | closedBall_compl_subset_of_mem_cocompact | null |
mem_cocompact_of_closedBall_compl_subset [ProperSpace α] (c : α)
(h : ∃ r, (closedBall c r)ᶜ ⊆ s) : s ∈ cocompact α := by
rcases h with ⟨r, h⟩
rw [Filter.mem_cocompact]
exact ⟨closedBall c r, isCompact_closedBall c r, h⟩ | theorem | Topology | [
"Mathlib.Topology.Order.Bornology",
"Mathlib.Topology.Order.Compact",
"Mathlib.Topology.MetricSpace.ProperSpace",
"Mathlib.Topology.MetricSpace.Cauchy",
"Mathlib.Topology.MetricSpace.Defs",
"Mathlib.Topology.EMetricSpace.Diam"
] | Mathlib/Topology/MetricSpace/Bounded.lean | mem_cocompact_of_closedBall_compl_subset | null |
mem_cocompact_iff_closedBall_compl_subset [ProperSpace α] (c : α) :
s ∈ cocompact α ↔ ∃ r, (closedBall c r)ᶜ ⊆ s :=
⟨(closedBall_compl_subset_of_mem_cocompact · _), mem_cocompact_of_closedBall_compl_subset _⟩ | theorem | Topology | [
"Mathlib.Topology.Order.Bornology",
"Mathlib.Topology.Order.Compact",
"Mathlib.Topology.MetricSpace.ProperSpace",
"Mathlib.Topology.MetricSpace.Cauchy",
"Mathlib.Topology.MetricSpace.Defs",
"Mathlib.Topology.EMetricSpace.Diam"
] | Mathlib/Topology/MetricSpace/Bounded.lean | mem_cocompact_iff_closedBall_compl_subset | null |
isBounded_range_iff {f : β → α} : IsBounded (range f) ↔ ∃ C, ∀ x y, dist (f x) (f y) ≤ C :=
isBounded_iff.trans <| by simp only [forall_mem_range] | theorem | Topology | [
"Mathlib.Topology.Order.Bornology",
"Mathlib.Topology.Order.Compact",
"Mathlib.Topology.MetricSpace.ProperSpace",
"Mathlib.Topology.MetricSpace.Cauchy",
"Mathlib.Topology.MetricSpace.Defs",
"Mathlib.Topology.EMetricSpace.Diam"
] | Mathlib/Topology/MetricSpace/Bounded.lean | isBounded_range_iff | Characterization of the boundedness of the range of a function |
isBounded_image_iff {f : β → α} {s : Set β} :
IsBounded (f '' s) ↔ ∃ C, ∀ x ∈ s, ∀ y ∈ s, dist (f x) (f y) ≤ C :=
isBounded_iff.trans <| by simp only [forall_mem_image] | theorem | Topology | [
"Mathlib.Topology.Order.Bornology",
"Mathlib.Topology.Order.Compact",
"Mathlib.Topology.MetricSpace.ProperSpace",
"Mathlib.Topology.MetricSpace.Cauchy",
"Mathlib.Topology.MetricSpace.Defs",
"Mathlib.Topology.EMetricSpace.Diam"
] | Mathlib/Topology/MetricSpace/Bounded.lean | isBounded_image_iff | null |
isBounded_range_of_tendsto_cofinite_uniformity {f : β → α}
(hf : Tendsto (Prod.map f f) (.cofinite ×ˢ .cofinite) (𝓤 α)) : IsBounded (range f) := by
rcases (hasBasis_cofinite.prod_self.tendsto_iff uniformity_basis_dist).1 hf 1 zero_lt_one with
⟨s, hsf, hs1⟩
rw [← image_union_image_compl_eq_range]
refine (... | theorem | Topology | [
"Mathlib.Topology.Order.Bornology",
"Mathlib.Topology.Order.Compact",
"Mathlib.Topology.MetricSpace.ProperSpace",
"Mathlib.Topology.MetricSpace.Cauchy",
"Mathlib.Topology.MetricSpace.Defs",
"Mathlib.Topology.EMetricSpace.Diam"
] | Mathlib/Topology/MetricSpace/Bounded.lean | isBounded_range_of_tendsto_cofinite_uniformity | null |
isBounded_range_of_cauchy_map_cofinite {f : β → α} (hf : Cauchy (map f cofinite)) :
IsBounded (range f) :=
isBounded_range_of_tendsto_cofinite_uniformity <| (cauchy_map_iff.1 hf).2 | theorem | Topology | [
"Mathlib.Topology.Order.Bornology",
"Mathlib.Topology.Order.Compact",
"Mathlib.Topology.MetricSpace.ProperSpace",
"Mathlib.Topology.MetricSpace.Cauchy",
"Mathlib.Topology.MetricSpace.Defs",
"Mathlib.Topology.EMetricSpace.Diam"
] | Mathlib/Topology/MetricSpace/Bounded.lean | isBounded_range_of_cauchy_map_cofinite | null |
_root_.CauchySeq.isBounded_range {f : ℕ → α} (hf : CauchySeq f) : IsBounded (range f) :=
isBounded_range_of_cauchy_map_cofinite <| by rwa [Nat.cofinite_eq_atTop] | theorem | Topology | [
"Mathlib.Topology.Order.Bornology",
"Mathlib.Topology.Order.Compact",
"Mathlib.Topology.MetricSpace.ProperSpace",
"Mathlib.Topology.MetricSpace.Cauchy",
"Mathlib.Topology.MetricSpace.Defs",
"Mathlib.Topology.EMetricSpace.Diam"
] | Mathlib/Topology/MetricSpace/Bounded.lean | _root_.CauchySeq.isBounded_range | null |
isBounded_range_of_tendsto_cofinite {f : β → α} {a : α} (hf : Tendsto f cofinite (𝓝 a)) :
IsBounded (range f) :=
isBounded_range_of_tendsto_cofinite_uniformity <|
(hf.prodMap hf).mono_right <| nhds_prod_eq.symm.trans_le (nhds_le_uniformity a) | theorem | Topology | [
"Mathlib.Topology.Order.Bornology",
"Mathlib.Topology.Order.Compact",
"Mathlib.Topology.MetricSpace.ProperSpace",
"Mathlib.Topology.MetricSpace.Cauchy",
"Mathlib.Topology.MetricSpace.Defs",
"Mathlib.Topology.EMetricSpace.Diam"
] | Mathlib/Topology/MetricSpace/Bounded.lean | isBounded_range_of_tendsto_cofinite | null |
isBounded_of_compactSpace [CompactSpace α] : IsBounded s :=
isCompact_univ.isBounded.subset (subset_univ _) | theorem | Topology | [
"Mathlib.Topology.Order.Bornology",
"Mathlib.Topology.Order.Compact",
"Mathlib.Topology.MetricSpace.ProperSpace",
"Mathlib.Topology.MetricSpace.Cauchy",
"Mathlib.Topology.MetricSpace.Defs",
"Mathlib.Topology.EMetricSpace.Diam"
] | Mathlib/Topology/MetricSpace/Bounded.lean | isBounded_of_compactSpace | In a compact space, all sets are bounded |
isBounded_range_of_tendsto (u : ℕ → α) {x : α} (hu : Tendsto u atTop (𝓝 x)) :
IsBounded (range u) :=
hu.cauchySeq.isBounded_range | theorem | Topology | [
"Mathlib.Topology.Order.Bornology",
"Mathlib.Topology.Order.Compact",
"Mathlib.Topology.MetricSpace.ProperSpace",
"Mathlib.Topology.MetricSpace.Cauchy",
"Mathlib.Topology.MetricSpace.Defs",
"Mathlib.Topology.EMetricSpace.Diam"
] | Mathlib/Topology/MetricSpace/Bounded.lean | isBounded_range_of_tendsto | null |
disjoint_nhds_cobounded (x : α) : Disjoint (𝓝 x) (cobounded α) :=
disjoint_of_disjoint_of_mem disjoint_compl_right (ball_mem_nhds _ one_pos) isBounded_ball | theorem | Topology | [
"Mathlib.Topology.Order.Bornology",
"Mathlib.Topology.Order.Compact",
"Mathlib.Topology.MetricSpace.ProperSpace",
"Mathlib.Topology.MetricSpace.Cauchy",
"Mathlib.Topology.MetricSpace.Defs",
"Mathlib.Topology.EMetricSpace.Diam"
] | Mathlib/Topology/MetricSpace/Bounded.lean | disjoint_nhds_cobounded | null |
disjoint_cobounded_nhds (x : α) : Disjoint (cobounded α) (𝓝 x) :=
(disjoint_nhds_cobounded x).symm | theorem | Topology | [
"Mathlib.Topology.Order.Bornology",
"Mathlib.Topology.Order.Compact",
"Mathlib.Topology.MetricSpace.ProperSpace",
"Mathlib.Topology.MetricSpace.Cauchy",
"Mathlib.Topology.MetricSpace.Defs",
"Mathlib.Topology.EMetricSpace.Diam"
] | Mathlib/Topology/MetricSpace/Bounded.lean | disjoint_cobounded_nhds | null |
disjoint_nhdsSet_cobounded {s : Set α} (hs : IsCompact s) : Disjoint (𝓝ˢ s) (cobounded α) :=
hs.disjoint_nhdsSet_left.2 fun _ _ ↦ disjoint_nhds_cobounded _ | theorem | Topology | [
"Mathlib.Topology.Order.Bornology",
"Mathlib.Topology.Order.Compact",
"Mathlib.Topology.MetricSpace.ProperSpace",
"Mathlib.Topology.MetricSpace.Cauchy",
"Mathlib.Topology.MetricSpace.Defs",
"Mathlib.Topology.EMetricSpace.Diam"
] | Mathlib/Topology/MetricSpace/Bounded.lean | disjoint_nhdsSet_cobounded | null |
disjoint_cobounded_nhdsSet {s : Set α} (hs : IsCompact s) : Disjoint (cobounded α) (𝓝ˢ s) :=
(disjoint_nhdsSet_cobounded hs).symm | theorem | Topology | [
"Mathlib.Topology.Order.Bornology",
"Mathlib.Topology.Order.Compact",
"Mathlib.Topology.MetricSpace.ProperSpace",
"Mathlib.Topology.MetricSpace.Cauchy",
"Mathlib.Topology.MetricSpace.Defs",
"Mathlib.Topology.EMetricSpace.Diam"
] | Mathlib/Topology/MetricSpace/Bounded.lean | disjoint_cobounded_nhdsSet | null |
exists_isBounded_image_of_tendsto {α β : Type*} [PseudoMetricSpace β]
{l : Filter α} {f : α → β} {x : β} (hf : Tendsto f l (𝓝 x)) :
∃ s ∈ l, IsBounded (f '' s) :=
(l.basis_sets.map f).disjoint_iff_left.mp <| (disjoint_nhds_cobounded x).mono_left hf | theorem | Topology | [
"Mathlib.Topology.Order.Bornology",
"Mathlib.Topology.Order.Compact",
"Mathlib.Topology.MetricSpace.ProperSpace",
"Mathlib.Topology.MetricSpace.Cauchy",
"Mathlib.Topology.MetricSpace.Defs",
"Mathlib.Topology.EMetricSpace.Diam"
] | Mathlib/Topology/MetricSpace/Bounded.lean | exists_isBounded_image_of_tendsto | null |
exists_isOpen_isBounded_image_inter_of_isCompact_of_forall_continuousWithinAt
[TopologicalSpace β] {k s : Set β} {f : β → α} (hk : IsCompact k)
(hf : ∀ x ∈ k, ContinuousWithinAt f s x) :
∃ t, k ⊆ t ∧ IsOpen t ∧ IsBounded (f '' (t ∩ s)) := by
have : Disjoint (𝓝ˢ k ⊓ 𝓟 s) (comap f (cobounded α)) := by
... | theorem | Topology | [
"Mathlib.Topology.Order.Bornology",
"Mathlib.Topology.Order.Compact",
"Mathlib.Topology.MetricSpace.ProperSpace",
"Mathlib.Topology.MetricSpace.Cauchy",
"Mathlib.Topology.MetricSpace.Defs",
"Mathlib.Topology.EMetricSpace.Diam"
] | Mathlib/Topology/MetricSpace/Bounded.lean | exists_isOpen_isBounded_image_inter_of_isCompact_of_forall_continuousWithinAt | If a function is continuous within a set `s` at every point of a compact set `k`, then it is
bounded on some open neighborhood of `k` in `s`. |
exists_isOpen_isBounded_image_of_isCompact_of_forall_continuousAt [TopologicalSpace β]
{k : Set β} {f : β → α} (hk : IsCompact k) (hf : ∀ x ∈ k, ContinuousAt f x) :
∃ t, k ⊆ t ∧ IsOpen t ∧ IsBounded (f '' t) := by
simp_rw [← continuousWithinAt_univ] at hf
simpa only [inter_univ] using
exists_isOpen_isBo... | theorem | Topology | [
"Mathlib.Topology.Order.Bornology",
"Mathlib.Topology.Order.Compact",
"Mathlib.Topology.MetricSpace.ProperSpace",
"Mathlib.Topology.MetricSpace.Cauchy",
"Mathlib.Topology.MetricSpace.Defs",
"Mathlib.Topology.EMetricSpace.Diam"
] | Mathlib/Topology/MetricSpace/Bounded.lean | exists_isOpen_isBounded_image_of_isCompact_of_forall_continuousAt | If a function is continuous at every point of a compact set `k`, then it is bounded on
some open neighborhood of `k`. |
exists_isOpen_isBounded_image_inter_of_isCompact_of_continuousOn [TopologicalSpace β]
{k s : Set β} {f : β → α} (hk : IsCompact k) (hks : k ⊆ s) (hf : ContinuousOn f s) :
∃ t, k ⊆ t ∧ IsOpen t ∧ IsBounded (f '' (t ∩ s)) :=
exists_isOpen_isBounded_image_inter_of_isCompact_of_forall_continuousWithinAt hk fun x ... | theorem | Topology | [
"Mathlib.Topology.Order.Bornology",
"Mathlib.Topology.Order.Compact",
"Mathlib.Topology.MetricSpace.ProperSpace",
"Mathlib.Topology.MetricSpace.Cauchy",
"Mathlib.Topology.MetricSpace.Defs",
"Mathlib.Topology.EMetricSpace.Diam"
] | Mathlib/Topology/MetricSpace/Bounded.lean | exists_isOpen_isBounded_image_inter_of_isCompact_of_continuousOn | If a function is continuous on a set `s` containing a compact set `k`, then it is bounded on
some open neighborhood of `k` in `s`. |
exists_isOpen_isBounded_image_of_isCompact_of_continuousOn [TopologicalSpace β]
{k s : Set β} {f : β → α} (hk : IsCompact k) (hs : IsOpen s) (hks : k ⊆ s)
(hf : ContinuousOn f s) : ∃ t, k ⊆ t ∧ IsOpen t ∧ IsBounded (f '' t) :=
exists_isOpen_isBounded_image_of_isCompact_of_forall_continuousAt hk fun _x hx =>
... | theorem | Topology | [
"Mathlib.Topology.Order.Bornology",
"Mathlib.Topology.Order.Compact",
"Mathlib.Topology.MetricSpace.ProperSpace",
"Mathlib.Topology.MetricSpace.Cauchy",
"Mathlib.Topology.MetricSpace.Defs",
"Mathlib.Topology.EMetricSpace.Diam"
] | Mathlib/Topology/MetricSpace/Bounded.lean | exists_isOpen_isBounded_image_of_isCompact_of_continuousOn | If a function is continuous on a neighborhood of a compact set `k`, then it is bounded on
some open neighborhood of `k`. |
isCompact_of_isClosed_isBounded [ProperSpace α] (hc : IsClosed s) (hb : IsBounded s) :
IsCompact s := by
rcases eq_empty_or_nonempty s with (rfl | ⟨x, -⟩)
· exact isCompact_empty
· rcases hb.subset_closedBall x with ⟨r, hr⟩
exact (isCompact_closedBall x r).of_isClosed_subset hc hr | theorem | Topology | [
"Mathlib.Topology.Order.Bornology",
"Mathlib.Topology.Order.Compact",
"Mathlib.Topology.MetricSpace.ProperSpace",
"Mathlib.Topology.MetricSpace.Cauchy",
"Mathlib.Topology.MetricSpace.Defs",
"Mathlib.Topology.EMetricSpace.Diam"
] | Mathlib/Topology/MetricSpace/Bounded.lean | isCompact_of_isClosed_isBounded | The **Heine–Borel theorem**: In a proper space, a closed bounded set is compact. |
_root_.Bornology.IsBounded.isCompact_closure [ProperSpace α] (h : IsBounded s) :
IsCompact (closure s) :=
isCompact_of_isClosed_isBounded isClosed_closure h.closure | theorem | Topology | [
"Mathlib.Topology.Order.Bornology",
"Mathlib.Topology.Order.Compact",
"Mathlib.Topology.MetricSpace.ProperSpace",
"Mathlib.Topology.MetricSpace.Cauchy",
"Mathlib.Topology.MetricSpace.Defs",
"Mathlib.Topology.EMetricSpace.Diam"
] | Mathlib/Topology/MetricSpace/Bounded.lean | _root_.Bornology.IsBounded.isCompact_closure | The **Heine–Borel theorem**: In a proper space, the closure of a bounded set is compact. |
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