fact stringlengths 6 3.84k | type stringclasses 11 values | library stringclasses 32 values | imports listlengths 1 14 | filename stringlengths 20 95 | symbolic_name stringlengths 1 90 | docstring stringlengths 7 20k ⌀ |
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@[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 [ENNReal.coe_mul, mul_assoc]; rfl | 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_maps y.mem⟩ | 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).trans_lt ?_
rw [mul_comm, ← div_eq_mul_inv] at hx
rw [mul_comm]
exact ENNReal.mul_lt_of_lt_div hx | 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
have B : IsBounded K := hK.isBounded_preimage isBounded_closedBall
have : IsCompact K := isCompact_iff_isClosed_bounded.2 ⟨A, B⟩
convert this.image f_cont
exact (hf.image_preimage _).symm | 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 s {b}) := by
intro h
apply hst.1
rw [Set.image2_singleton_right] at h
replace h := (hf b).isBounded_preimage h
exact h.subset (subset_preimage_image _ _)
exact mt (IsBounded.subset · (image2_subset subset_rfl (singleton_subset_iff.mpr hb))) this | 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, uniformContinuous_iff] | 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_uniformity hε) <| by simpa using hs
theorem isClosedEmbedding_of_pairwise_le_dist {α : Type*} [TopologicalSpace α] [DiscreteTopology α]
{ε : ℝ} (hε : 0 < ε) {f : α → γ} (hf : Pairwise fun x y => ε ≤ dist (f x) (f y)) :
IsClosedEmbedding f :=
isClosedEmbedding_of_spaced_out (dist_mem_uniformity hε) <| by simpa using hf
/-- If `f : β → α` sends any two distinct points to points at distance at least `ε > 0`, then
`f` is a uniform embedding with respect to the discrete uniformity on `β`. |
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 which is not defeq to the push-forward
of the edistance to reals. |
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⟩
let Finv := rangeSplitting F
refine ⟨range Finv, ⟨countable_range _, fun x => ?_⟩⟩
let x' := Finv ⟨F x, mem_range_self _⟩
have : F x' = F x := apply_rangeSplitting F _
exact ⟨x', mem_range_self _, hF _ _ this.symm⟩ | 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 synonyms.
Important Note: if `α` is some synonym of a type `β` (at default transparency), and `f : α ≃ β` is
some bilipschitz equivalence, then instead of writing:
```
instance : UniformSpace α := inferInstanceAs (UniformSpace β)
```
Users should instead write something like:
```
instance : UniformSpace α := (inferInstance : UniformSpace β).comap f
```
in order to avoid abuse of the definitional equality `α := β`. |
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 synonyms.
Important Note: if `α` is some synonym of a type `β` (at default transparency), and `f : α ≃ β` is
some bilipschitz equivalence, then instead of writing:
```
instance : Bornology α := inferInstanceAs (Bornology β)
```
Users should instead write something like:
```
instance : Bornology α := Bornology.induced (f : α → β)
```
in order to avoid abuse of the definitional equality `α := β`. |
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 (hsf.image f).isBounded.union (isBounded_image_iff.2 ⟨1, fun x hx y hy ↦ ?_⟩)
exact le_of_lt (hs1 (x, y) ⟨hx, hy⟩) | 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
rw [disjoint_assoc, inf_comm, hk.disjoint_nhdsSet_left]
exact fun x hx ↦ disjoint_left_comm.2 <|
tendsto_comap.disjoint (disjoint_cobounded_nhds _) (hf x hx)
rcases ((((hasBasis_nhdsSet _).inf_principal _)).disjoint_iff ((basis_sets _).comap _)).1 this
with ⟨U, ⟨hUo, hkU⟩, t, ht, hd⟩
refine ⟨U, hkU, hUo, (isBounded_compl_iff.2 ht).subset ?_⟩
rwa [image_subset_iff, preimage_compl, subset_compl_iff_disjoint_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 | 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_isBounded_image_inter_of_isCompact_of_forall_continuousWithinAt hk 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_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 hx =>
hf x (hks 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_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 =>
hf.continuousAt (hs.mem_nhds (hks 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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