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 ⌀ |
|---|---|---|---|---|---|---|
nhds_infty_eq : 𝓝 (∞ : OnePoint X) = map (↑) (coclosedCompact X) ⊔ pure ∞ := by
rw [← nhdsNE_infty_eq, nhdsNE_sup_pure] | theorem | Topology | [
"Mathlib.Data.Fintype.Option",
"Mathlib.Topology.Homeomorph.Lemmas",
"Mathlib.Topology.Sets.Opens"
] | Mathlib/Topology/Compactification/OnePoint/Basic.lean | nhds_infty_eq | null |
tendsto_coe_infty : Tendsto (↑) (coclosedCompact X) (𝓝 (∞ : OnePoint X)) := by
rw [nhds_infty_eq]
exact Filter.Tendsto.mono_right tendsto_map le_sup_left | theorem | Topology | [
"Mathlib.Data.Fintype.Option",
"Mathlib.Topology.Homeomorph.Lemmas",
"Mathlib.Topology.Sets.Opens"
] | Mathlib/Topology/Compactification/OnePoint/Basic.lean | tendsto_coe_infty | null |
hasBasis_nhds_infty :
(𝓝 (∞ : OnePoint X)).HasBasis (fun s : Set X => IsClosed s ∧ IsCompact s) fun s =>
(↑) '' sᶜ ∪ {∞} := by
rw [nhds_infty_eq]
exact (hasBasis_coclosedCompact.map _).sup_pure _
@[simp] | theorem | Topology | [
"Mathlib.Data.Fintype.Option",
"Mathlib.Topology.Homeomorph.Lemmas",
"Mathlib.Topology.Sets.Opens"
] | Mathlib/Topology/Compactification/OnePoint/Basic.lean | hasBasis_nhds_infty | null |
comap_coe_nhds_infty : comap ((↑) : X → OnePoint X) (𝓝 ∞) = coclosedCompact X := by
simp [nhds_infty_eq, comap_sup, comap_map coe_injective] | theorem | Topology | [
"Mathlib.Data.Fintype.Option",
"Mathlib.Topology.Homeomorph.Lemmas",
"Mathlib.Topology.Sets.Opens"
] | Mathlib/Topology/Compactification/OnePoint/Basic.lean | comap_coe_nhds_infty | null |
le_nhds_infty {f : Filter (OnePoint X)} :
f ≤ 𝓝 ∞ ↔ ∀ s : Set X, IsClosed s → IsCompact s → (↑) '' sᶜ ∪ {∞} ∈ f := by
simp only [hasBasis_nhds_infty.ge_iff, and_imp] | theorem | Topology | [
"Mathlib.Data.Fintype.Option",
"Mathlib.Topology.Homeomorph.Lemmas",
"Mathlib.Topology.Sets.Opens"
] | Mathlib/Topology/Compactification/OnePoint/Basic.lean | le_nhds_infty | null |
ultrafilter_le_nhds_infty {f : Ultrafilter (OnePoint X)} :
(f : Filter (OnePoint X)) ≤ 𝓝 ∞ ↔ ∀ s : Set X, IsClosed s → IsCompact s → (↑) '' s ∉ f := by
simp only [le_nhds_infty, ← compl_image_coe, Ultrafilter.mem_coe,
Ultrafilter.compl_mem_iff_notMem] | theorem | Topology | [
"Mathlib.Data.Fintype.Option",
"Mathlib.Topology.Homeomorph.Lemmas",
"Mathlib.Topology.Sets.Opens"
] | Mathlib/Topology/Compactification/OnePoint/Basic.lean | ultrafilter_le_nhds_infty | null |
tendsto_nhds_infty' {α : Type*} {f : OnePoint X → α} {l : Filter α} :
Tendsto f (𝓝 ∞) l ↔ Tendsto f (pure ∞) l ∧ Tendsto (f ∘ (↑)) (coclosedCompact X) l := by
simp [nhds_infty_eq, and_comm] | theorem | Topology | [
"Mathlib.Data.Fintype.Option",
"Mathlib.Topology.Homeomorph.Lemmas",
"Mathlib.Topology.Sets.Opens"
] | Mathlib/Topology/Compactification/OnePoint/Basic.lean | tendsto_nhds_infty' | null |
tendsto_nhds_infty {α : Type*} {f : OnePoint X → α} {l : Filter α} :
Tendsto f (𝓝 ∞) l ↔
∀ s ∈ l, f ∞ ∈ s ∧ ∃ t : Set X, IsClosed t ∧ IsCompact t ∧ MapsTo (f ∘ (↑)) tᶜ s :=
tendsto_nhds_infty'.trans <| by
simp only [tendsto_pure_left, hasBasis_coclosedCompact.tendsto_left_iff, forall_and,
and_ass... | theorem | Topology | [
"Mathlib.Data.Fintype.Option",
"Mathlib.Topology.Homeomorph.Lemmas",
"Mathlib.Topology.Sets.Opens"
] | Mathlib/Topology/Compactification/OnePoint/Basic.lean | tendsto_nhds_infty | null |
continuousAt_infty' {Y : Type*} [TopologicalSpace Y] {f : OnePoint X → Y} :
ContinuousAt f ∞ ↔ Tendsto (f ∘ (↑)) (coclosedCompact X) (𝓝 (f ∞)) :=
tendsto_nhds_infty'.trans <| and_iff_right (tendsto_pure_nhds _ _) | theorem | Topology | [
"Mathlib.Data.Fintype.Option",
"Mathlib.Topology.Homeomorph.Lemmas",
"Mathlib.Topology.Sets.Opens"
] | Mathlib/Topology/Compactification/OnePoint/Basic.lean | continuousAt_infty' | null |
continuousAt_infty {Y : Type*} [TopologicalSpace Y] {f : OnePoint X → Y} :
ContinuousAt f ∞ ↔
∀ s ∈ 𝓝 (f ∞), ∃ t : Set X, IsClosed t ∧ IsCompact t ∧ MapsTo (f ∘ (↑)) tᶜ s :=
continuousAt_infty'.trans <| by simp only [hasBasis_coclosedCompact.tendsto_left_iff, and_assoc] | theorem | Topology | [
"Mathlib.Data.Fintype.Option",
"Mathlib.Topology.Homeomorph.Lemmas",
"Mathlib.Topology.Sets.Opens"
] | Mathlib/Topology/Compactification/OnePoint/Basic.lean | continuousAt_infty | null |
continuousAt_coe {Y : Type*} [TopologicalSpace Y] {f : OnePoint X → Y} {x : X} :
ContinuousAt f x ↔ ContinuousAt (f ∘ (↑)) x := by
rw [ContinuousAt, nhds_coe_eq, tendsto_map'_iff, ContinuousAt]; rfl | theorem | Topology | [
"Mathlib.Data.Fintype.Option",
"Mathlib.Topology.Homeomorph.Lemmas",
"Mathlib.Topology.Sets.Opens"
] | Mathlib/Topology/Compactification/OnePoint/Basic.lean | continuousAt_coe | null |
continuous_iff {Y : Type*} [TopologicalSpace Y] (f : OnePoint X → Y) : Continuous f ↔
Tendsto (fun x : X ↦ f x) (coclosedCompact X) (𝓝 (f ∞)) ∧ Continuous (fun x : X ↦ f x) := by
simp only [continuous_iff_continuousAt, OnePoint.forall, continuousAt_coe, continuousAt_infty',
Function.comp_def] | lemma | Topology | [
"Mathlib.Data.Fintype.Option",
"Mathlib.Topology.Homeomorph.Lemmas",
"Mathlib.Topology.Sets.Opens"
] | Mathlib/Topology/Compactification/OnePoint/Basic.lean | continuous_iff | null |
continuousMapMk {Y : Type*} [TopologicalSpace Y] (f : C(X, Y)) (y : Y)
(h : Tendsto f (coclosedCompact X) (𝓝 y)) : C(OnePoint X, Y) where
toFun x := x.elim y f
continuous_toFun := by
rw [continuous_iff]
refine ⟨h, f.continuous⟩ | def | Topology | [
"Mathlib.Data.Fintype.Option",
"Mathlib.Topology.Homeomorph.Lemmas",
"Mathlib.Topology.Sets.Opens"
] | Mathlib/Topology/Compactification/OnePoint/Basic.lean | continuousMapMk | A constructor for continuous maps out of a one point compactification, given a continuous map from
the underlying space and a limit value at infinity. |
continuous_iff_from_discrete {Y : Type*} [TopologicalSpace Y]
[DiscreteTopology X] (f : OnePoint X → Y) :
Continuous f ↔ Tendsto (fun x : X ↦ f x) cofinite (𝓝 (f ∞)) := by
simp [continuous_iff, cocompact_eq_cofinite, continuous_of_discreteTopology] | lemma | Topology | [
"Mathlib.Data.Fintype.Option",
"Mathlib.Topology.Homeomorph.Lemmas",
"Mathlib.Topology.Sets.Opens"
] | Mathlib/Topology/Compactification/OnePoint/Basic.lean | continuous_iff_from_discrete | null |
continuousMapMkDiscrete {Y : Type*} [TopologicalSpace Y]
[DiscreteTopology X] (f : X → Y) (y : Y) (h : Tendsto f cofinite (𝓝 y)) :
C(OnePoint X, Y) :=
continuousMapMk ⟨f, continuous_of_discreteTopology⟩ y (by simpa [cocompact_eq_cofinite])
variable (X) in | def | Topology | [
"Mathlib.Data.Fintype.Option",
"Mathlib.Topology.Homeomorph.Lemmas",
"Mathlib.Topology.Sets.Opens"
] | Mathlib/Topology/Compactification/OnePoint/Basic.lean | continuousMapMkDiscrete | A constructor for continuous maps out of a one point compactification of a discrete space, given a
map from the underlying space and a limit value at infinity. |
noncomputable continuousMapDiscreteEquiv (Y : Type*) [DiscreteTopology X] [TopologicalSpace Y]
[T2Space Y] [Infinite X] :
C(OnePoint X, Y) ≃ { f : X → Y // ∃ L, Tendsto (fun x : X ↦ f x) cofinite (𝓝 L) } where
toFun f := ⟨(f ·), ⟨f ∞, continuous_iff_from_discrete _ |>.mp (map_continuous f)⟩⟩
invFun f :=
... | def | Topology | [
"Mathlib.Data.Fintype.Option",
"Mathlib.Topology.Homeomorph.Lemmas",
"Mathlib.Topology.Sets.Opens"
] | Mathlib/Topology/Compactification/OnePoint/Basic.lean | continuousMapDiscreteEquiv | Continuous maps out of the one point compactification of an infinite discrete space to a Hausdorff
space correspond bijectively to "convergent" maps out of the discrete space. |
continuous_iff_from_nat {Y : Type*} [TopologicalSpace Y] (f : OnePoint ℕ → Y) :
Continuous f ↔ Tendsto (fun x : ℕ ↦ f x) atTop (𝓝 (f ∞)) := by
rw [continuous_iff_from_discrete, Nat.cofinite_eq_atTop] | lemma | Topology | [
"Mathlib.Data.Fintype.Option",
"Mathlib.Topology.Homeomorph.Lemmas",
"Mathlib.Topology.Sets.Opens"
] | Mathlib/Topology/Compactification/OnePoint/Basic.lean | continuous_iff_from_nat | null |
continuousMapMkNat {Y : Type*} [TopologicalSpace Y]
(f : ℕ → Y) (y : Y) (h : Tendsto f atTop (𝓝 y)) :
C(OnePoint ℕ, Y) :=
continuousMapMkDiscrete f y (by rwa [Nat.cofinite_eq_atTop]) | def | Topology | [
"Mathlib.Data.Fintype.Option",
"Mathlib.Topology.Homeomorph.Lemmas",
"Mathlib.Topology.Sets.Opens"
] | Mathlib/Topology/Compactification/OnePoint/Basic.lean | continuousMapMkNat | A constructor for continuous maps out of the one point compactification of `ℕ`, given a
sequence and a limit value at infinity. |
noncomputable continuousMapNatEquiv (Y : Type*) [TopologicalSpace Y] [T2Space Y] :
C(OnePoint ℕ, Y) ≃ { f : ℕ → Y // ∃ L, Tendsto (f ·) atTop (𝓝 L) } := by
refine (continuousMapDiscreteEquiv ℕ Y).trans {
toFun := fun ⟨f, hf⟩ ↦ ⟨f, by rwa [← Nat.cofinite_eq_atTop]⟩
invFun := fun ⟨f, hf⟩ ↦ ⟨f, by rwa [Nat.... | def | Topology | [
"Mathlib.Data.Fintype.Option",
"Mathlib.Topology.Homeomorph.Lemmas",
"Mathlib.Topology.Sets.Opens"
] | Mathlib/Topology/Compactification/OnePoint/Basic.lean | continuousMapNatEquiv | Continuous maps out of the one point compactification of `ℕ` to a Hausdorff space `Y` correspond
bijectively to convergent sequences in `Y`. |
denseRange_coe [NoncompactSpace X] : DenseRange ((↑) : X → OnePoint X) := by
rw [DenseRange, ← compl_infty]
exact dense_compl_singleton _ | theorem | Topology | [
"Mathlib.Data.Fintype.Option",
"Mathlib.Topology.Homeomorph.Lemmas",
"Mathlib.Topology.Sets.Opens"
] | Mathlib/Topology/Compactification/OnePoint/Basic.lean | denseRange_coe | If `X` is not a compact space, then the natural embedding `X → OnePoint X` has dense range. |
isDenseEmbedding_coe [NoncompactSpace X] : IsDenseEmbedding ((↑) : X → OnePoint X) :=
{ isOpenEmbedding_coe with dense := denseRange_coe }
@[simp, norm_cast] | theorem | Topology | [
"Mathlib.Data.Fintype.Option",
"Mathlib.Topology.Homeomorph.Lemmas",
"Mathlib.Topology.Sets.Opens"
] | Mathlib/Topology/Compactification/OnePoint/Basic.lean | isDenseEmbedding_coe | null |
specializes_coe {x y : X} : (x : OnePoint X) ⤳ y ↔ x ⤳ y :=
isOpenEmbedding_coe.isInducing.specializes_iff
@[simp, norm_cast] | theorem | Topology | [
"Mathlib.Data.Fintype.Option",
"Mathlib.Topology.Homeomorph.Lemmas",
"Mathlib.Topology.Sets.Opens"
] | Mathlib/Topology/Compactification/OnePoint/Basic.lean | specializes_coe | null |
inseparable_coe {x y : X} : Inseparable (x : OnePoint X) y ↔ Inseparable x y :=
isOpenEmbedding_coe.isInducing.inseparable_iff | theorem | Topology | [
"Mathlib.Data.Fintype.Option",
"Mathlib.Topology.Homeomorph.Lemmas",
"Mathlib.Topology.Sets.Opens"
] | Mathlib/Topology/Compactification/OnePoint/Basic.lean | inseparable_coe | null |
not_specializes_infty_coe {x : X} : ¬Specializes ∞ (x : OnePoint X) :=
isClosed_infty.not_specializes rfl (coe_ne_infty x) | theorem | Topology | [
"Mathlib.Data.Fintype.Option",
"Mathlib.Topology.Homeomorph.Lemmas",
"Mathlib.Topology.Sets.Opens"
] | Mathlib/Topology/Compactification/OnePoint/Basic.lean | not_specializes_infty_coe | null |
not_inseparable_infty_coe {x : X} : ¬Inseparable ∞ (x : OnePoint X) := fun h =>
not_specializes_infty_coe h.specializes | theorem | Topology | [
"Mathlib.Data.Fintype.Option",
"Mathlib.Topology.Homeomorph.Lemmas",
"Mathlib.Topology.Sets.Opens"
] | Mathlib/Topology/Compactification/OnePoint/Basic.lean | not_inseparable_infty_coe | null |
not_inseparable_coe_infty {x : X} : ¬Inseparable (x : OnePoint X) ∞ := fun h =>
not_specializes_infty_coe h.specializes' | theorem | Topology | [
"Mathlib.Data.Fintype.Option",
"Mathlib.Topology.Homeomorph.Lemmas",
"Mathlib.Topology.Sets.Opens"
] | Mathlib/Topology/Compactification/OnePoint/Basic.lean | not_inseparable_coe_infty | null |
inseparable_iff {x y : OnePoint X} :
Inseparable x y ↔ x = ∞ ∧ y = ∞ ∨ ∃ x' : X, x = x' ∧ ∃ y' : X, y = y' ∧ Inseparable x' y' := by
induction x using OnePoint.rec <;> induction y using OnePoint.rec <;>
simp [not_inseparable_infty_coe, not_inseparable_coe_infty, coe_eq_coe, Inseparable.refl] | theorem | Topology | [
"Mathlib.Data.Fintype.Option",
"Mathlib.Topology.Homeomorph.Lemmas",
"Mathlib.Topology.Sets.Opens"
] | Mathlib/Topology/Compactification/OnePoint/Basic.lean | inseparable_iff | null |
continuous_map_iff [TopologicalSpace Y] {f : X → Y} :
Continuous (OnePoint.map f) ↔
Continuous f ∧ Tendsto f (coclosedCompact X) (coclosedCompact Y) := by
simp_rw [continuous_iff, map_some, ← comap_coe_nhds_infty, tendsto_comap_iff, map_infty,
isOpenEmbedding_coe.isInducing.continuous_iff (Y := Y)]
ex... | theorem | Topology | [
"Mathlib.Data.Fintype.Option",
"Mathlib.Topology.Homeomorph.Lemmas",
"Mathlib.Topology.Sets.Opens"
] | Mathlib/Topology/Compactification/OnePoint/Basic.lean | continuous_map_iff | null |
continuous_map [TopologicalSpace Y] {f : X → Y} (hc : Continuous f)
(h : Tendsto f (coclosedCompact X) (coclosedCompact Y)) :
Continuous (OnePoint.map f) :=
continuous_map_iff.mpr ⟨hc, h⟩
/-! | theorem | Topology | [
"Mathlib.Data.Fintype.Option",
"Mathlib.Topology.Homeomorph.Lemmas",
"Mathlib.Topology.Sets.Opens"
] | Mathlib/Topology/Compactification/OnePoint/Basic.lean | continuous_map | null |
not_continuous_cofiniteTopology_of_symm [Infinite X] [DiscreteTopology X] :
¬Continuous (@CofiniteTopology.of (OnePoint X)).symm := by
inhabit X
simp only [continuous_iff_continuousAt, ContinuousAt, not_forall]
use CofiniteTopology.of ↑(default : X)
simpa [nhds_coe_eq, nhds_discrete, CofiniteTopology.nhds_e... | theorem | Topology | [
"Mathlib.Data.Fintype.Option",
"Mathlib.Topology.Homeomorph.Lemmas",
"Mathlib.Topology.Sets.Opens"
] | Mathlib/Topology/Compactification/OnePoint/Basic.lean | not_continuous_cofiniteTopology_of_symm | For any topological space `X`, its one point compactification is a compact space. -/
instance : CompactSpace (OnePoint X) where
isCompact_univ := by
have : Tendsto ((↑) : X → OnePoint X) (cocompact X) (𝓝 ∞) := by
rw [nhds_infty_eq]
exact (tendsto_map.mono_left cocompact_le_coclosedCompact).mono_right... |
noncomputable equivOfIsEmbeddingOfRangeEq :
OnePoint X ≃ₜ Y :=
have _i := hf.t2Space
have : Tendsto f (coclosedCompact X) (𝓝 y) := by
rw [coclosedCompact_eq_cocompact, hasBasis_cocompact.tendsto_left_iff]
intro N hN
obtain ⟨U, hU₁, hU₂, hU₃⟩ := mem_nhds_iff.mp hN
refine ⟨f⁻¹' Uᶜ, ?_, by simpa u... | def | Topology | [
"Mathlib.Data.Fintype.Option",
"Mathlib.Topology.Homeomorph.Lemmas",
"Mathlib.Topology.Sets.Opens"
] | Mathlib/Topology/Compactification/OnePoint/Basic.lean | equivOfIsEmbeddingOfRangeEq | If `f` embeds `X` into a compact Hausdorff space `Y`, and has exactly one point outside its
range, then `(Y, f)` is the one-point compactification of `X`. |
equivOfIsEmbeddingOfRangeEq_apply_coe (x : X) :
equivOfIsEmbeddingOfRangeEq y f hf hy x = f x :=
rfl
@[simp] | lemma | Topology | [
"Mathlib.Data.Fintype.Option",
"Mathlib.Topology.Homeomorph.Lemmas",
"Mathlib.Topology.Sets.Opens"
] | Mathlib/Topology/Compactification/OnePoint/Basic.lean | equivOfIsEmbeddingOfRangeEq_apply_coe | null |
equivOfIsEmbeddingOfRangeEq_apply_infty :
equivOfIsEmbeddingOfRangeEq y f hf hy ∞ = y :=
rfl | lemma | Topology | [
"Mathlib.Data.Fintype.Option",
"Mathlib.Topology.Homeomorph.Lemmas",
"Mathlib.Topology.Sets.Opens"
] | Mathlib/Topology/Compactification/OnePoint/Basic.lean | equivOfIsEmbeddingOfRangeEq_apply_infty | null |
@[simps]
onePointCongr (h : X ≃ₜ Y) : OnePoint X ≃ₜ OnePoint Y where
__ := h.toEquiv.withTopCongr
toFun := OnePoint.map h
invFun := OnePoint.map h.symm
continuous_toFun := continuous_map (map_continuous h) h.map_coclosedCompact.le
continuous_invFun := continuous_map (map_continuous h.symm) h.symm.map_coclosed... | def | Topology | [
"Mathlib.Data.Fintype.Option",
"Mathlib.Topology.Homeomorph.Lemmas",
"Mathlib.Topology.Sets.Opens"
] | Mathlib/Topology/Compactification/OnePoint/Basic.lean | onePointCongr | Extend a homeomorphism of topological spaces
to the homeomorphism of their one point compactifications. |
Continuous.homeoOfEquivCompactToT2.t1_counterexample :
∃ (α β : Type) (_ : TopologicalSpace α) (_ : TopologicalSpace β),
CompactSpace α ∧ T1Space β ∧ ∃ f : α ≃ β, Continuous f ∧ ¬Continuous f.symm :=
⟨OnePoint ℕ, CofiniteTopology (OnePoint ℕ), inferInstance, inferInstance, inferInstance,
inferInstance, ... | theorem | Topology | [
"Mathlib.Data.Fintype.Option",
"Mathlib.Topology.Homeomorph.Lemmas",
"Mathlib.Topology.Sets.Opens"
] | Mathlib/Topology/Compactification/OnePoint/Basic.lean | Continuous.homeoOfEquivCompactToT2.t1_counterexample | A concrete counterexample shows that `Continuous.homeoOfEquivCompactToT2`
cannot be generalized from `T2Space` to `T1Space`.
Let `α = OnePoint ℕ` be the one-point compactification of `ℕ`, and let `β` be the same space
`OnePoint ℕ` with the cofinite topology. Then `α` is compact, `β` is T1, and the identity map
`id : ... |
@[simp] Matrix.fin_two_smul_prod (g : Matrix (Fin 2) (Fin 2) R) (v : R × R) :
g • v = (g 0 0 * v.1 + g 0 1 * v.2, g 1 0 * v.1 + g 1 1 * v.2) := by
simp [Equiv.smul_def, smul_eq_mulVec, Matrix.mulVec_eq_sum]
@[simp] lemma Matrix.GeneralLinearGroup.fin_two_smul_prod {R : Type*} [CommRing R]
(g : GL (Fin 2) R) (... | lemma | Topology | [
"Mathlib.Algebra.QuadraticDiscriminant",
"Mathlib.Data.Matrix.Action",
"Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup.FinTwo",
"Mathlib.LinearAlgebra.Projectivization.Action",
"Mathlib.Topology.Compactification.OnePoint.Basic"
] | Mathlib/Topology/Compactification/OnePoint/ProjectiveLine.lean | Matrix.fin_two_smul_prod | null |
equivProjectivization :
OnePoint K ≃ ℙ K (K × K) where
toFun p := p.elim (mk K (1, 0) (by simp)) (fun t ↦ mk K (t, 1) (by simp))
invFun p := by
refine Projectivization.lift
(fun u : {v : K × K // v ≠ 0} ↦ if u.1.2 = 0 then ∞ else ((u.1.2)⁻¹ * u.1.1)) ?_ p
rintro ⟨-, hv⟩ ⟨⟨x, y⟩, hw⟩ t rfl
have... | def | Topology | [
"Mathlib.Algebra.QuadraticDiscriminant",
"Mathlib.Data.Matrix.Action",
"Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup.FinTwo",
"Mathlib.LinearAlgebra.Projectivization.Action",
"Mathlib.Topology.Compactification.OnePoint.Basic"
] | Mathlib/Topology/Compactification/OnePoint/ProjectiveLine.lean | equivProjectivization | The one-point compactification of a division ring `K` is equivalent to
the projectivization `ℙ K (K × K)`. |
equivProjectivization_apply_infinity :
equivProjectivization K ∞ = mk K ⟨1, 0⟩ (by simp) :=
rfl
@[simp] | lemma | Topology | [
"Mathlib.Algebra.QuadraticDiscriminant",
"Mathlib.Data.Matrix.Action",
"Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup.FinTwo",
"Mathlib.LinearAlgebra.Projectivization.Action",
"Mathlib.Topology.Compactification.OnePoint.Basic"
] | Mathlib/Topology/Compactification/OnePoint/ProjectiveLine.lean | equivProjectivization_apply_infinity | null |
equivProjectivization_apply_coe (t : K) :
equivProjectivization K t = mk K ⟨t, 1⟩ (by simp) :=
rfl
@[simp] | lemma | Topology | [
"Mathlib.Algebra.QuadraticDiscriminant",
"Mathlib.Data.Matrix.Action",
"Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup.FinTwo",
"Mathlib.LinearAlgebra.Projectivization.Action",
"Mathlib.Topology.Compactification.OnePoint.Basic"
] | Mathlib/Topology/Compactification/OnePoint/ProjectiveLine.lean | equivProjectivization_apply_coe | null |
equivProjectivization_symm_apply_mk (x y : K) (h : (x, y) ≠ 0) :
(equivProjectivization K).symm (mk K ⟨x, y⟩ h) = if y = 0 then ∞ else y⁻¹ * x := by
simp [equivProjectivization] | lemma | Topology | [
"Mathlib.Algebra.QuadraticDiscriminant",
"Mathlib.Data.Matrix.Action",
"Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup.FinTwo",
"Mathlib.LinearAlgebra.Projectivization.Action",
"Mathlib.Topology.Compactification.OnePoint.Basic"
] | Mathlib/Topology/Compactification/OnePoint/ProjectiveLine.lean | equivProjectivization_symm_apply_mk | null |
instGLAction : MulAction (GL (Fin 2) K) (OnePoint K) :=
(equivProjectivization K).mulAction (GL (Fin 2) K) | instance | Topology | [
"Mathlib.Algebra.QuadraticDiscriminant",
"Mathlib.Data.Matrix.Action",
"Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup.FinTwo",
"Mathlib.LinearAlgebra.Projectivization.Action",
"Mathlib.Topology.Compactification.OnePoint.Basic"
] | Mathlib/Topology/Compactification/OnePoint/ProjectiveLine.lean | instGLAction | For a field `K`, the group `GL(2, K)` acts on `OnePoint K`, via the canonical identification
with the `ℙ¹(K)` (which is given explicitly by Möbius transformations). |
smul_infty_def {g : GL (Fin 2) K} :
g • ∞ = (equivProjectivization K).symm (.mk K (g 0 0, g 1 0) (fun h ↦ by
simpa [det_fin_two, Prod.mk_eq_zero.mp h] using g.det_ne_zero)) := by
simp [Equiv.smul_def, mulVec_eq_sum, Units.smul_def] | lemma | Topology | [
"Mathlib.Algebra.QuadraticDiscriminant",
"Mathlib.Data.Matrix.Action",
"Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup.FinTwo",
"Mathlib.LinearAlgebra.Projectivization.Action",
"Mathlib.Topology.Compactification.OnePoint.Basic"
] | Mathlib/Topology/Compactification/OnePoint/ProjectiveLine.lean | smul_infty_def | null |
smul_infty_eq_ite (g : GL (Fin 2) K) :
g • (∞ : OnePoint K) = if g 1 0 = 0 then ∞ else g 0 0 / g 1 0 := by
by_cases h : g 1 0 = 0 <;>
simp [h, div_eq_inv_mul, smul_infty_def] | lemma | Topology | [
"Mathlib.Algebra.QuadraticDiscriminant",
"Mathlib.Data.Matrix.Action",
"Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup.FinTwo",
"Mathlib.LinearAlgebra.Projectivization.Action",
"Mathlib.Topology.Compactification.OnePoint.Basic"
] | Mathlib/Topology/Compactification/OnePoint/ProjectiveLine.lean | smul_infty_eq_ite | null |
smul_infty_eq_self_iff {g : GL (Fin 2) K} :
g • (∞ : OnePoint K) = ∞ ↔ g 1 0 = 0 := by
simp [smul_infty_eq_ite] | lemma | Topology | [
"Mathlib.Algebra.QuadraticDiscriminant",
"Mathlib.Data.Matrix.Action",
"Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup.FinTwo",
"Mathlib.LinearAlgebra.Projectivization.Action",
"Mathlib.Topology.Compactification.OnePoint.Basic"
] | Mathlib/Topology/Compactification/OnePoint/ProjectiveLine.lean | smul_infty_eq_self_iff | null |
smul_some_eq_ite {g : GL (Fin 2) K} {k : K} :
g • (k : OnePoint K) =
if g 1 0 * k + g 1 1 = 0 then ∞ else (g 0 0 * k + g 0 1) / (g 1 0 * k + g 1 1) := by
simp [Equiv.smul_def, mulVec_eq_sum, div_eq_inv_mul, mul_comm, Units.smul_def] | lemma | Topology | [
"Mathlib.Algebra.QuadraticDiscriminant",
"Mathlib.Data.Matrix.Action",
"Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup.FinTwo",
"Mathlib.LinearAlgebra.Projectivization.Action",
"Mathlib.Topology.Compactification.OnePoint.Basic"
] | Mathlib/Topology/Compactification/OnePoint/ProjectiveLine.lean | smul_some_eq_ite | null |
map_smul {L : Type*} [Field L] [DecidableEq L]
(f : K →+* L) (g : GL (Fin 2) K) (c : OnePoint K) :
OnePoint.map f (g • c) = (g.map f) • (c.map f) := by
cases c with
| infty => simp [smul_infty_eq_ite, apply_ite]
| coe c => simp [smul_some_eq_ite, ← map_mul, ← map_add, apply_ite] | lemma | Topology | [
"Mathlib.Algebra.QuadraticDiscriminant",
"Mathlib.Data.Matrix.Action",
"Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup.FinTwo",
"Mathlib.LinearAlgebra.Projectivization.Action",
"Mathlib.Topology.Compactification.OnePoint.Basic"
] | Mathlib/Topology/Compactification/OnePoint/ProjectiveLine.lean | map_smul | null |
fixpointPolynomial_aeval_eq_zero_iff {c : K} {g : GL (Fin 2) K} :
g.fixpointPolynomial.aeval c = 0 ↔ g • (c : OnePoint K) = c := by
simp only [fixpointPolynomial, map_sub, map_mul, map_add, aeval_X_pow, aeval_C, aeval_X,
Algebra.algebraMap_self_apply, OnePoint.smul_some_eq_ite]
split_ifs with h
· refine ⟨... | lemma | Topology | [
"Mathlib.Algebra.QuadraticDiscriminant",
"Mathlib.Data.Matrix.Action",
"Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup.FinTwo",
"Mathlib.LinearAlgebra.Projectivization.Action",
"Mathlib.Topology.Compactification.OnePoint.Basic"
] | Mathlib/Topology/Compactification/OnePoint/ProjectiveLine.lean | fixpointPolynomial_aeval_eq_zero_iff | The roots of `g.fixpointPolynomial` are the fixed points of `g ∈ GL(2, K)` acting on the finite
part of `OnePoint K`. |
parabolicFixedPoint (g : GL (Fin 2) K) : OnePoint K :=
if g 1 0 = 0 then ∞ else ↑((g 0 0 - g 1 1) / (2 * g 1 0)) | def | Topology | [
"Mathlib.Algebra.QuadraticDiscriminant",
"Mathlib.Data.Matrix.Action",
"Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup.FinTwo",
"Mathlib.LinearAlgebra.Projectivization.Action",
"Mathlib.Topology.Compactification.OnePoint.Basic"
] | Mathlib/Topology/Compactification/OnePoint/ProjectiveLine.lean | parabolicFixedPoint | If `g` is parabolic, this is the unique fixed point of `g` in `OnePoint K`. |
IsParabolic.smul_eq_self_iff {g : GL (Fin 2) K} (hg : g.IsParabolic) [NeZero (2 : K)]
{c : OnePoint K} : g • c = c ↔ c = parabolicFixedPoint g := by
rcases hg with ⟨hg, hdisc⟩
rw [disc_fin_two, trace_fin_two, det_fin_two] at hdisc
cases c with
| infty => by_cases h : g 1 0 = 0 <;> simp [parabolicFixedPoint,... | lemma | Topology | [
"Mathlib.Algebra.QuadraticDiscriminant",
"Mathlib.Data.Matrix.Action",
"Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup.FinTwo",
"Mathlib.LinearAlgebra.Projectivization.Action",
"Mathlib.Topology.Compactification.OnePoint.Basic"
] | Mathlib/Topology/Compactification/OnePoint/ProjectiveLine.lean | IsParabolic.smul_eq_self_iff | null |
IsParabolic.parabolicFixedPoint_pow {g : GL (Fin 2) K} (hg : IsParabolic g) [CharZero K]
{n : ℕ} (hn : n ≠ 0) :
(g ^ n).parabolicFixedPoint = g.parabolicFixedPoint := by
rw [eq_comm, ← IsParabolic.smul_eq_self_iff (hg.pow hn)]
clear hn
induction n with
| zero => simp
| succ n IH => rw [pow_succ, MulAc... | lemma | Topology | [
"Mathlib.Algebra.QuadraticDiscriminant",
"Mathlib.Data.Matrix.Action",
"Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup.FinTwo",
"Mathlib.LinearAlgebra.Projectivization.Action",
"Mathlib.Topology.Compactification.OnePoint.Basic"
] | Mathlib/Topology/Compactification/OnePoint/ProjectiveLine.lean | IsParabolic.parabolicFixedPoint_pow | null |
IsElliptic.smul_ne_self [LinearOrder K] [IsStrictOrderedRing K]
{g : GL (Fin 2) K} (hg : g.IsElliptic) (c : OnePoint K) :
g • c ≠ c := by
cases c with
| infty =>
rw [Ne, smul_infty_eq_self_iff]
refine fun h ↦ not_le_of_gt hg ?_
have : g.val.disc = (g 0 0 - g 1 1) ^ 2 := by
simp only [disc_... | lemma | Topology | [
"Mathlib.Algebra.QuadraticDiscriminant",
"Mathlib.Data.Matrix.Action",
"Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup.FinTwo",
"Mathlib.LinearAlgebra.Projectivization.Action",
"Mathlib.Topology.Compactification.OnePoint.Basic"
] | Mathlib/Topology/Compactification/OnePoint/ProjectiveLine.lean | IsElliptic.smul_ne_self | Elliptic elements have no fixed points in `OnePoint K`. |
onePointHyperplaneHomeoUnitSphere
{E : Type*} [NormedAddCommGroup E] [InnerProductSpace ℝ E] [FiniteDimensional ℝ E]
{v : E} (hv : ‖v‖ = 1) :
OnePoint (ℝ ∙ v)ᗮ ≃ₜ sphere (0 : E) 1 :=
OnePoint.equivOfIsEmbeddingOfRangeEq _ _
(isOpenEmbedding_stereographic_symm hv).toIsEmbedding (range_stereographic_sym... | def | Topology | [
"Mathlib.Topology.Compactification.OnePoint.Basic",
"Mathlib.Geometry.Manifold.Instances.Sphere"
] | Mathlib/Topology/Compactification/OnePoint/Sphere.lean | onePointHyperplaneHomeoUnitSphere | A homeomorphism from the one-point compactification of a hyperplane in Euclidean space to the
sphere. |
onePointEquivSphereOfFinrankEq {ι V : Type*} [Fintype ι]
[AddCommGroup V] [Module ℝ V] [FiniteDimensional ℝ V]
[TopologicalSpace V] [IsTopologicalAddGroup V] [ContinuousSMul ℝ V] [T2Space V]
(h : finrank ℝ V + 1 = Fintype.card ι) :
OnePoint V ≃ₜ sphere (0 : EuclideanSpace ℝ ι) 1 := by
classical
have... | def | Topology | [
"Mathlib.Topology.Compactification.OnePoint.Basic",
"Mathlib.Geometry.Manifold.Instances.Sphere"
] | Mathlib/Topology/Compactification/OnePoint/Sphere.lean | onePointEquivSphereOfFinrankEq | A homeomorphism from the one-point compactification of a finite-dimensional real vector space to
the sphere. |
arzela_ascoli₁ [CompactSpace β] (A : Set (α →ᵇ β)) (closed : IsClosed A)
(H : Equicontinuous ((↑) : A → α → β)) : IsCompact A := by
simp_rw [Equicontinuous, Metric.equicontinuousAt_iff_pair] at H
refine TotallyBounded.isCompact_of_isClosed ?_ closed
refine totallyBounded_of_finite_discretization fun ε ε0 => ?... | theorem | Topology | [
"Mathlib.Topology.ContinuousMap.Bounded.Basic",
"Mathlib.Topology.MetricSpace.Equicontinuity"
] | Mathlib/Topology/ContinuousMap/Bounded/ArzelaAscoli.lean | arzela_ascoli₁ | First version, with pointwise equicontinuity and range in a compact space. |
arzela_ascoli₂ (s : Set β) (hs : IsCompact s) (A : Set (α →ᵇ β)) (closed : IsClosed A)
(in_s : ∀ (f : α →ᵇ β) (x : α), f ∈ A → f x ∈ s) (H : Equicontinuous ((↑) : A → α → β)) :
IsCompact A := by
/- This version is deduced from the previous one by restricting to the compact type in the target,
using compactn... | theorem | Topology | [
"Mathlib.Topology.ContinuousMap.Bounded.Basic",
"Mathlib.Topology.MetricSpace.Equicontinuity"
] | Mathlib/Topology/ContinuousMap/Bounded/ArzelaAscoli.lean | arzela_ascoli₂ | Second version, with pointwise equicontinuity and range in a compact subset. |
arzela_ascoli [T2Space β] (s : Set β) (hs : IsCompact s) (A : Set (α →ᵇ β))
(in_s : ∀ (f : α →ᵇ β) (x : α), f ∈ A → f x ∈ s) (H : Equicontinuous ((↑) : A → α → β)) :
IsCompact (closure A) :=
/- This version is deduced from the previous one by checking that the closure of `A`, in
addition to being closed, st... | theorem | Topology | [
"Mathlib.Topology.ContinuousMap.Bounded.Basic",
"Mathlib.Topology.MetricSpace.Equicontinuity"
] | Mathlib/Topology/ContinuousMap/Bounded/ArzelaAscoli.lean | arzela_ascoli | Third (main) version, with pointwise equicontinuity and range in a compact subset, but
without closedness. The closure is then compact. |
BoundedContinuousFunction (α : Type u) (β : Type v) [TopologicalSpace α]
[PseudoMetricSpace β] : Type max u v extends ContinuousMap α β where
map_bounded' : ∃ C, ∀ x y, dist (toFun x) (toFun y) ≤ C
@[inherit_doc] scoped[BoundedContinuousFunction] infixr:25 " →ᵇ " => BoundedContinuousFunction | structure | Topology | [
"Mathlib.Topology.Algebra.Indicator",
"Mathlib.Topology.Bornology.BoundedOperation",
"Mathlib.Topology.ContinuousMap.Algebra"
] | Mathlib/Topology/ContinuousMap/Bounded/Basic.lean | BoundedContinuousFunction | `α →ᵇ β` is the type of bounded continuous functions `α → β` from a topological space to a
metric space.
When possible, instead of parametrizing results over `(f : α →ᵇ β)`,
you should parametrize over `(F : Type*) [BoundedContinuousMapClass F α β] (f : F)`.
When you extend this structure, make sure to extend `Bounde... |
BoundedContinuousMapClass (F : Type*) (α β : outParam Type*) [TopologicalSpace α]
[PseudoMetricSpace β] [FunLike F α β] : Prop extends ContinuousMapClass F α β where
map_bounded (f : F) : ∃ C, ∀ x y, dist (f x) (f y) ≤ C | class | Topology | [
"Mathlib.Topology.Algebra.Indicator",
"Mathlib.Topology.Bornology.BoundedOperation",
"Mathlib.Topology.ContinuousMap.Algebra"
] | Mathlib/Topology/ContinuousMap/Bounded/Basic.lean | BoundedContinuousMapClass | `BoundedContinuousMapClass F α β` states that `F` is a type of bounded continuous maps.
You should also extend this typeclass when you extend `BoundedContinuousFunction`. |
instFunLike : FunLike (α →ᵇ β) α β where
coe f := f.toFun
coe_injective' f g h := by
obtain ⟨⟨_, _⟩, _⟩ := f
obtain ⟨⟨_, _⟩, _⟩ := g
congr | instance | Topology | [
"Mathlib.Topology.Algebra.Indicator",
"Mathlib.Topology.Bornology.BoundedOperation",
"Mathlib.Topology.ContinuousMap.Algebra"
] | Mathlib/Topology/ContinuousMap/Bounded/Basic.lean | instFunLike | null |
instBoundedContinuousMapClass : BoundedContinuousMapClass (α →ᵇ β) α β where
map_continuous f := f.continuous_toFun
map_bounded f := f.map_bounded' | instance | Topology | [
"Mathlib.Topology.Algebra.Indicator",
"Mathlib.Topology.Bornology.BoundedOperation",
"Mathlib.Topology.ContinuousMap.Algebra"
] | Mathlib/Topology/ContinuousMap/Bounded/Basic.lean | instBoundedContinuousMapClass | null |
instCoeTC [FunLike F α β] [BoundedContinuousMapClass F α β] : CoeTC F (α →ᵇ β) :=
⟨fun f =>
{ toFun := f
continuous_toFun := map_continuous f
map_bounded' := map_bounded f }⟩
@[simp] | instance | Topology | [
"Mathlib.Topology.Algebra.Indicator",
"Mathlib.Topology.Bornology.BoundedOperation",
"Mathlib.Topology.ContinuousMap.Algebra"
] | Mathlib/Topology/ContinuousMap/Bounded/Basic.lean | instCoeTC | null |
coe_toContinuousMap (f : α →ᵇ β) : (f.toContinuousMap : α → β) = f := rfl | theorem | Topology | [
"Mathlib.Topology.Algebra.Indicator",
"Mathlib.Topology.Bornology.BoundedOperation",
"Mathlib.Topology.ContinuousMap.Algebra"
] | Mathlib/Topology/ContinuousMap/Bounded/Basic.lean | coe_toContinuousMap | null |
Simps.apply (h : α →ᵇ β) : α → β := h
initialize_simps_projections BoundedContinuousFunction (toFun → apply) | def | Topology | [
"Mathlib.Topology.Algebra.Indicator",
"Mathlib.Topology.Bornology.BoundedOperation",
"Mathlib.Topology.ContinuousMap.Algebra"
] | Mathlib/Topology/ContinuousMap/Bounded/Basic.lean | Simps.apply | See Note [custom simps projection]. We need to specify this projection explicitly in this case,
because it is a composition of multiple projections. |
protected bounded (f : α →ᵇ β) : ∃ C, ∀ x y : α, dist (f x) (f y) ≤ C :=
f.map_bounded' | theorem | Topology | [
"Mathlib.Topology.Algebra.Indicator",
"Mathlib.Topology.Bornology.BoundedOperation",
"Mathlib.Topology.ContinuousMap.Algebra"
] | Mathlib/Topology/ContinuousMap/Bounded/Basic.lean | bounded | null |
protected continuous (f : α →ᵇ β) : Continuous f :=
f.toContinuousMap.continuous
@[ext] | theorem | Topology | [
"Mathlib.Topology.Algebra.Indicator",
"Mathlib.Topology.Bornology.BoundedOperation",
"Mathlib.Topology.ContinuousMap.Algebra"
] | Mathlib/Topology/ContinuousMap/Bounded/Basic.lean | continuous | null |
ext (h : ∀ x, f x = g x) : f = g :=
DFunLike.ext _ _ h | theorem | Topology | [
"Mathlib.Topology.Algebra.Indicator",
"Mathlib.Topology.Bornology.BoundedOperation",
"Mathlib.Topology.ContinuousMap.Algebra"
] | Mathlib/Topology/ContinuousMap/Bounded/Basic.lean | ext | null |
isBounded_range (f : α →ᵇ β) : IsBounded (range f) :=
isBounded_range_iff.2 f.bounded | theorem | Topology | [
"Mathlib.Topology.Algebra.Indicator",
"Mathlib.Topology.Bornology.BoundedOperation",
"Mathlib.Topology.ContinuousMap.Algebra"
] | Mathlib/Topology/ContinuousMap/Bounded/Basic.lean | isBounded_range | null |
isBounded_image (f : α →ᵇ β) (s : Set α) : IsBounded (f '' s) :=
f.isBounded_range.subset <| image_subset_range _ _ | theorem | Topology | [
"Mathlib.Topology.Algebra.Indicator",
"Mathlib.Topology.Bornology.BoundedOperation",
"Mathlib.Topology.ContinuousMap.Algebra"
] | Mathlib/Topology/ContinuousMap/Bounded/Basic.lean | isBounded_image | null |
eq_of_empty [h : IsEmpty α] (f g : α →ᵇ β) : f = g :=
ext <| h.elim | theorem | Topology | [
"Mathlib.Topology.Algebra.Indicator",
"Mathlib.Topology.Bornology.BoundedOperation",
"Mathlib.Topology.ContinuousMap.Algebra"
] | Mathlib/Topology/ContinuousMap/Bounded/Basic.lean | eq_of_empty | null |
mkOfBound (f : C(α, β)) (C : ℝ) (h : ∀ x y : α, dist (f x) (f y) ≤ C) : α →ᵇ β :=
⟨f, ⟨C, h⟩⟩
@[simp] | def | Topology | [
"Mathlib.Topology.Algebra.Indicator",
"Mathlib.Topology.Bornology.BoundedOperation",
"Mathlib.Topology.ContinuousMap.Algebra"
] | Mathlib/Topology/ContinuousMap/Bounded/Basic.lean | mkOfBound | A continuous function with an explicit bound is a bounded continuous function. |
mkOfBound_coe {f} {C} {h} : (mkOfBound f C h : α → β) = (f : α → β) := rfl | theorem | Topology | [
"Mathlib.Topology.Algebra.Indicator",
"Mathlib.Topology.Bornology.BoundedOperation",
"Mathlib.Topology.ContinuousMap.Algebra"
] | Mathlib/Topology/ContinuousMap/Bounded/Basic.lean | mkOfBound_coe | null |
mkOfCompact [CompactSpace α] (f : C(α, β)) : α →ᵇ β :=
⟨f, isBounded_range_iff.1 (isCompact_range f.continuous).isBounded⟩
@[simp] | def | Topology | [
"Mathlib.Topology.Algebra.Indicator",
"Mathlib.Topology.Bornology.BoundedOperation",
"Mathlib.Topology.ContinuousMap.Algebra"
] | Mathlib/Topology/ContinuousMap/Bounded/Basic.lean | mkOfCompact | A continuous function on a compact space is automatically a bounded continuous function. |
mkOfCompact_apply [CompactSpace α] (f : C(α, β)) (a : α) : mkOfCompact f a = f a := rfl | theorem | Topology | [
"Mathlib.Topology.Algebra.Indicator",
"Mathlib.Topology.Bornology.BoundedOperation",
"Mathlib.Topology.ContinuousMap.Algebra"
] | Mathlib/Topology/ContinuousMap/Bounded/Basic.lean | mkOfCompact_apply | null |
@[simps]
mkOfDiscrete [DiscreteTopology α] (f : α → β) (C : ℝ) (h : ∀ x y : α, dist (f x) (f y) ≤ C) :
α →ᵇ β :=
⟨⟨f, continuous_of_discreteTopology⟩, ⟨C, h⟩⟩ | def | Topology | [
"Mathlib.Topology.Algebra.Indicator",
"Mathlib.Topology.Bornology.BoundedOperation",
"Mathlib.Topology.ContinuousMap.Algebra"
] | Mathlib/Topology/ContinuousMap/Bounded/Basic.lean | mkOfDiscrete | If a function is bounded on a discrete space, it is automatically continuous,
and therefore gives rise to an element of the type of bounded continuous functions. |
instDist : Dist (α →ᵇ β) :=
⟨fun f g => sInf { C | 0 ≤ C ∧ ∀ x : α, dist (f x) (g x) ≤ C }⟩ | instance | Topology | [
"Mathlib.Topology.Algebra.Indicator",
"Mathlib.Topology.Bornology.BoundedOperation",
"Mathlib.Topology.ContinuousMap.Algebra"
] | Mathlib/Topology/ContinuousMap/Bounded/Basic.lean | instDist | The uniform distance between two bounded continuous functions. |
dist_eq : dist f g = sInf { C | 0 ≤ C ∧ ∀ x : α, dist (f x) (g x) ≤ C } := rfl | theorem | Topology | [
"Mathlib.Topology.Algebra.Indicator",
"Mathlib.Topology.Bornology.BoundedOperation",
"Mathlib.Topology.ContinuousMap.Algebra"
] | Mathlib/Topology/ContinuousMap/Bounded/Basic.lean | dist_eq | null |
dist_set_exists : ∃ C, 0 ≤ C ∧ ∀ x : α, dist (f x) (g x) ≤ C := by
rcases isBounded_iff.1 (f.isBounded_range.union g.isBounded_range) with ⟨C, hC⟩
refine ⟨max 0 C, le_max_left _ _, fun x => (hC ?_ ?_).trans (le_max_right _ _)⟩
<;> [left; right]
<;> apply mem_range_self | theorem | Topology | [
"Mathlib.Topology.Algebra.Indicator",
"Mathlib.Topology.Bornology.BoundedOperation",
"Mathlib.Topology.ContinuousMap.Algebra"
] | Mathlib/Topology/ContinuousMap/Bounded/Basic.lean | dist_set_exists | null |
dist_coe_le_dist (x : α) : dist (f x) (g x) ≤ dist f g :=
le_csInf dist_set_exists fun _ hb => hb.2 x
/- This lemma will be needed in the proof of the metric space instance, but it will become
useless afterwards as it will be superseded by the general result that the distance is nonnegative
in metric spaces. -/ | theorem | Topology | [
"Mathlib.Topology.Algebra.Indicator",
"Mathlib.Topology.Bornology.BoundedOperation",
"Mathlib.Topology.ContinuousMap.Algebra"
] | Mathlib/Topology/ContinuousMap/Bounded/Basic.lean | dist_coe_le_dist | The pointwise distance is controlled by the distance between functions, by definition. |
private dist_nonneg' : 0 ≤ dist f g :=
le_csInf dist_set_exists fun _ => And.left | theorem | Topology | [
"Mathlib.Topology.Algebra.Indicator",
"Mathlib.Topology.Bornology.BoundedOperation",
"Mathlib.Topology.ContinuousMap.Algebra"
] | Mathlib/Topology/ContinuousMap/Bounded/Basic.lean | dist_nonneg' | null |
dist_le (C0 : (0 : ℝ) ≤ C) : dist f g ≤ C ↔ ∀ x : α, dist (f x) (g x) ≤ C :=
⟨fun h x => le_trans (dist_coe_le_dist x) h, fun H => csInf_le ⟨0, fun _ => And.left⟩ ⟨C0, H⟩⟩ | theorem | Topology | [
"Mathlib.Topology.Algebra.Indicator",
"Mathlib.Topology.Bornology.BoundedOperation",
"Mathlib.Topology.ContinuousMap.Algebra"
] | Mathlib/Topology/ContinuousMap/Bounded/Basic.lean | dist_le | The distance between two functions is controlled by the supremum of the pointwise distances. |
dist_le_iff_of_nonempty [Nonempty α] : dist f g ≤ C ↔ ∀ x, dist (f x) (g x) ≤ C :=
⟨fun h x => le_trans (dist_coe_le_dist x) h,
fun w => (dist_le (le_trans dist_nonneg (w (Nonempty.some ‹_›)))).mpr w⟩ | theorem | Topology | [
"Mathlib.Topology.Algebra.Indicator",
"Mathlib.Topology.Bornology.BoundedOperation",
"Mathlib.Topology.ContinuousMap.Algebra"
] | Mathlib/Topology/ContinuousMap/Bounded/Basic.lean | dist_le_iff_of_nonempty | null |
dist_lt_of_nonempty_compact [Nonempty α] [CompactSpace α]
(w : ∀ x : α, dist (f x) (g x) < C) : dist f g < C := by
have c : Continuous fun x => dist (f x) (g x) := by fun_prop
obtain ⟨x, -, le⟩ :=
IsCompact.exists_isMaxOn isCompact_univ Set.univ_nonempty (Continuous.continuousOn c)
exact lt_of_le_of_lt (d... | theorem | Topology | [
"Mathlib.Topology.Algebra.Indicator",
"Mathlib.Topology.Bornology.BoundedOperation",
"Mathlib.Topology.ContinuousMap.Algebra"
] | Mathlib/Topology/ContinuousMap/Bounded/Basic.lean | dist_lt_of_nonempty_compact | null |
dist_lt_iff_of_compact [CompactSpace α] (C0 : (0 : ℝ) < C) :
dist f g < C ↔ ∀ x : α, dist (f x) (g x) < C := by
fconstructor
· intro w x
exact lt_of_le_of_lt (dist_coe_le_dist x) w
· by_cases h : Nonempty α
· exact dist_lt_of_nonempty_compact
· rintro -
convert C0
apply le_antisymm _ d... | theorem | Topology | [
"Mathlib.Topology.Algebra.Indicator",
"Mathlib.Topology.Bornology.BoundedOperation",
"Mathlib.Topology.ContinuousMap.Algebra"
] | Mathlib/Topology/ContinuousMap/Bounded/Basic.lean | dist_lt_iff_of_compact | null |
dist_lt_iff_of_nonempty_compact [Nonempty α] [CompactSpace α] :
dist f g < C ↔ ∀ x : α, dist (f x) (g x) < C :=
⟨fun w x => lt_of_le_of_lt (dist_coe_le_dist x) w, dist_lt_of_nonempty_compact⟩ | theorem | Topology | [
"Mathlib.Topology.Algebra.Indicator",
"Mathlib.Topology.Bornology.BoundedOperation",
"Mathlib.Topology.ContinuousMap.Algebra"
] | Mathlib/Topology/ContinuousMap/Bounded/Basic.lean | dist_lt_iff_of_nonempty_compact | null |
instPseudoMetricSpace : PseudoMetricSpace (α →ᵇ β) where
dist_self f := le_antisymm ((dist_le le_rfl).2 fun x => by simp) dist_nonneg'
dist_comm f g := by simp [dist_eq, dist_comm]
dist_triangle _ _ _ := (dist_le (add_nonneg dist_nonneg' dist_nonneg')).2
fun _ => le_trans (dist_triangle _ _ _) (add_le_add (di... | instance | Topology | [
"Mathlib.Topology.Algebra.Indicator",
"Mathlib.Topology.Bornology.BoundedOperation",
"Mathlib.Topology.ContinuousMap.Algebra"
] | Mathlib/Topology/ContinuousMap/Bounded/Basic.lean | instPseudoMetricSpace | The type of bounded continuous functions, with the uniform distance, is a pseudometric space. |
instMetricSpace {β} [MetricSpace β] : MetricSpace (α →ᵇ β) where
eq_of_dist_eq_zero hfg := by
ext x
exact eq_of_dist_eq_zero (le_antisymm (hfg ▸ dist_coe_le_dist _) dist_nonneg) | instance | Topology | [
"Mathlib.Topology.Algebra.Indicator",
"Mathlib.Topology.Bornology.BoundedOperation",
"Mathlib.Topology.ContinuousMap.Algebra"
] | Mathlib/Topology/ContinuousMap/Bounded/Basic.lean | instMetricSpace | The type of bounded continuous functions, with the uniform distance, is a metric space. |
nndist_eq : nndist f g = sInf { C | ∀ x : α, nndist (f x) (g x) ≤ C } :=
Subtype.ext <| dist_eq.trans <| by
rw [val_eq_coe, coe_sInf, coe_image]
simp_rw [mem_setOf_eq, ← NNReal.coe_le_coe, coe_mk, exists_prop, coe_nndist] | theorem | Topology | [
"Mathlib.Topology.Algebra.Indicator",
"Mathlib.Topology.Bornology.BoundedOperation",
"Mathlib.Topology.ContinuousMap.Algebra"
] | Mathlib/Topology/ContinuousMap/Bounded/Basic.lean | nndist_eq | null |
nndist_set_exists : ∃ C, ∀ x : α, nndist (f x) (g x) ≤ C :=
Subtype.exists.mpr <| dist_set_exists.imp fun _ ⟨ha, h⟩ => ⟨ha, h⟩ | theorem | Topology | [
"Mathlib.Topology.Algebra.Indicator",
"Mathlib.Topology.Bornology.BoundedOperation",
"Mathlib.Topology.ContinuousMap.Algebra"
] | Mathlib/Topology/ContinuousMap/Bounded/Basic.lean | nndist_set_exists | null |
nndist_coe_le_nndist (x : α) : nndist (f x) (g x) ≤ nndist f g :=
dist_coe_le_dist x | theorem | Topology | [
"Mathlib.Topology.Algebra.Indicator",
"Mathlib.Topology.Bornology.BoundedOperation",
"Mathlib.Topology.ContinuousMap.Algebra"
] | Mathlib/Topology/ContinuousMap/Bounded/Basic.lean | nndist_coe_le_nndist | null |
dist_zero_of_empty [IsEmpty α] : dist f g = 0 := by
rw [(ext isEmptyElim : f = g), dist_self] | theorem | Topology | [
"Mathlib.Topology.Algebra.Indicator",
"Mathlib.Topology.Bornology.BoundedOperation",
"Mathlib.Topology.ContinuousMap.Algebra"
] | Mathlib/Topology/ContinuousMap/Bounded/Basic.lean | dist_zero_of_empty | On an empty space, bounded continuous functions are at distance 0. |
dist_eq_iSup : dist f g = ⨆ x : α, dist (f x) (g x) := by
cases isEmpty_or_nonempty α
· rw [iSup_of_empty', Real.sSup_empty, dist_zero_of_empty]
refine (dist_le_iff_of_nonempty.mpr <| le_ciSup ?_).antisymm (ciSup_le dist_coe_le_dist)
exact dist_set_exists.imp fun C hC => forall_mem_range.2 hC.2 | theorem | Topology | [
"Mathlib.Topology.Algebra.Indicator",
"Mathlib.Topology.Bornology.BoundedOperation",
"Mathlib.Topology.ContinuousMap.Algebra"
] | Mathlib/Topology/ContinuousMap/Bounded/Basic.lean | dist_eq_iSup | null |
nndist_eq_iSup : nndist f g = ⨆ x : α, nndist (f x) (g x) :=
Subtype.ext <| dist_eq_iSup.trans <| by simp_rw [val_eq_coe, coe_iSup, coe_nndist] | theorem | Topology | [
"Mathlib.Topology.Algebra.Indicator",
"Mathlib.Topology.Bornology.BoundedOperation",
"Mathlib.Topology.ContinuousMap.Algebra"
] | Mathlib/Topology/ContinuousMap/Bounded/Basic.lean | nndist_eq_iSup | null |
edist_eq_iSup : edist f g = ⨆ x, edist (f x) (g x) := by
simp_rw [edist_nndist, nndist_eq_iSup]
refine ENNReal.coe_iSup ⟨nndist f g, ?_⟩
rintro - ⟨x, hx, rfl⟩
exact nndist_coe_le_nndist x | theorem | Topology | [
"Mathlib.Topology.Algebra.Indicator",
"Mathlib.Topology.Bornology.BoundedOperation",
"Mathlib.Topology.ContinuousMap.Algebra"
] | Mathlib/Topology/ContinuousMap/Bounded/Basic.lean | edist_eq_iSup | null |
tendsto_iff_tendstoUniformly {ι : Type*} {F : ι → α →ᵇ β} {f : α →ᵇ β} {l : Filter ι} :
Tendsto F l (𝓝 f) ↔ TendstoUniformly (fun i => F i) f l :=
Iff.intro
(fun h =>
tendstoUniformly_iff.2 fun ε ε0 =>
(Metric.tendsto_nhds.mp h ε ε0).mp
(Eventually.of_forall fun n hn x =>
... | theorem | Topology | [
"Mathlib.Topology.Algebra.Indicator",
"Mathlib.Topology.Bornology.BoundedOperation",
"Mathlib.Topology.ContinuousMap.Algebra"
] | Mathlib/Topology/ContinuousMap/Bounded/Basic.lean | tendsto_iff_tendstoUniformly | null |
isInducing_coeFn : IsInducing (UniformFun.ofFun ∘ (⇑) : (α →ᵇ β) → α →ᵤ β) := by
rw [isInducing_iff_nhds]
refine fun f => eq_of_forall_le_iff fun l => ?_
rw [← tendsto_iff_comap, ← tendsto_id', tendsto_iff_tendstoUniformly,
UniformFun.tendsto_iff_tendstoUniformly]
simp [comp_def] | theorem | Topology | [
"Mathlib.Topology.Algebra.Indicator",
"Mathlib.Topology.Bornology.BoundedOperation",
"Mathlib.Topology.ContinuousMap.Algebra"
] | Mathlib/Topology/ContinuousMap/Bounded/Basic.lean | isInducing_coeFn | The topology on `α →ᵇ β` is exactly the topology induced by the natural map to `α →ᵤ β`. |
isEmbedding_coeFn : IsEmbedding (UniformFun.ofFun ∘ (⇑) : (α →ᵇ β) → α →ᵤ β) :=
⟨isInducing_coeFn, fun _ _ h => ext fun x => congr_fun h x⟩
variable (α) in | theorem | Topology | [
"Mathlib.Topology.Algebra.Indicator",
"Mathlib.Topology.Bornology.BoundedOperation",
"Mathlib.Topology.ContinuousMap.Algebra"
] | Mathlib/Topology/ContinuousMap/Bounded/Basic.lean | isEmbedding_coeFn | null |
@[simps! -fullyApplied]
const (b : β) : α →ᵇ β :=
⟨ContinuousMap.const α b, 0, by simp⟩ | def | Topology | [
"Mathlib.Topology.Algebra.Indicator",
"Mathlib.Topology.Bornology.BoundedOperation",
"Mathlib.Topology.ContinuousMap.Algebra"
] | Mathlib/Topology/ContinuousMap/Bounded/Basic.lean | const | Constant as a continuous bounded function. |
const_apply' (a : α) (b : β) : (const α b : α → β) a = b := rfl | theorem | Topology | [
"Mathlib.Topology.Algebra.Indicator",
"Mathlib.Topology.Bornology.BoundedOperation",
"Mathlib.Topology.ContinuousMap.Algebra"
] | Mathlib/Topology/ContinuousMap/Bounded/Basic.lean | const_apply' | null |
@[continuity]
continuous_eval_const {x : α} : Continuous fun f : α →ᵇ β => f x :=
(continuous_apply x).comp continuous_coe | theorem | Topology | [
"Mathlib.Topology.Algebra.Indicator",
"Mathlib.Topology.Bornology.BoundedOperation",
"Mathlib.Topology.ContinuousMap.Algebra"
] | Mathlib/Topology/ContinuousMap/Bounded/Basic.lean | continuous_eval_const | If the target space is inhabited, so is the space of bounded continuous functions. -/
instance [Inhabited β] : Inhabited (α →ᵇ β) :=
⟨const α default⟩
theorem lipschitz_evalx (x : α) : LipschitzWith 1 fun f : α →ᵇ β => f x :=
LipschitzWith.mk_one fun _ _ => dist_coe_le_dist x
theorem uniformContinuous_coe : @Unif... |
@[continuity]
continuous_eval : Continuous fun p : (α →ᵇ β) × α => p.1 p.2 :=
(continuous_prod_of_continuous_lipschitzWith _ 1 fun f => f.continuous) <| lipschitz_evalx | theorem | Topology | [
"Mathlib.Topology.Algebra.Indicator",
"Mathlib.Topology.Bornology.BoundedOperation",
"Mathlib.Topology.ContinuousMap.Algebra"
] | Mathlib/Topology/ContinuousMap/Bounded/Basic.lean | continuous_eval | The evaluation map is continuous, as a joint function of `u` and `x`. |
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