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 ⌀ |
|---|---|---|---|---|---|---|
_root_.Fin.appendHomeomorph_toEquiv (m n : ℕ) :
(Fin.appendHomeomorph (X := X) m n).toEquiv = Fin.appendEquiv m n :=
rfl | theorem | Topology | [
"Mathlib.Logic.Equiv.Fin.Basic",
"Mathlib.Topology.Connected.LocallyConnected",
"Mathlib.Topology.DenseEmbedding",
"Mathlib.Topology.Connected.TotallyDisconnected"
] | Mathlib/Topology/Homeomorph/Lemmas.lean | _root_.Fin.appendHomeomorph_toEquiv | null |
@[simps! apply symm_apply toEquiv]
sigmaProdDistrib : (Σ i, X i) × Y ≃ₜ Σ i, X i × Y :=
Homeomorph.symm <|
(Equiv.sigmaProdDistrib X Y).symm.toHomeomorphOfContinuousOpen
(continuous_sigma fun _ => continuous_sigmaMk.fst'.prodMk continuous_snd)
(isOpenMap_sigma.2 fun _ => isOpenMap_sigmaMk.prodMap IsOpenMap.id) | def | Topology | [
"Mathlib.Logic.Equiv.Fin.Basic",
"Mathlib.Topology.Connected.LocallyConnected",
"Mathlib.Topology.DenseEmbedding",
"Mathlib.Topology.Connected.TotallyDisconnected"
] | Mathlib/Topology/Homeomorph/Lemmas.lean | sigmaProdDistrib | `(Σ i, X i) × Y` is homeomorphic to `Σ i, (X i × Y)`. |
@[simps! -fullyApplied]
funUnique (ι X : Type*) [Unique ι] [TopologicalSpace X] : (ι → X) ≃ₜ X where
toEquiv := Equiv.funUnique ι X
continuous_toFun := continuous_apply _
continuous_invFun := continuous_pi fun _ => continuous_id | def | Topology | [
"Mathlib.Logic.Equiv.Fin.Basic",
"Mathlib.Topology.Connected.LocallyConnected",
"Mathlib.Topology.DenseEmbedding",
"Mathlib.Topology.Connected.TotallyDisconnected"
] | Mathlib/Topology/Homeomorph/Lemmas.lean | funUnique | If `ι` has a unique element, then `ι → X` is homeomorphic to `X`. |
@[simps! -fullyApplied]
piFinTwo.{u} (X : Fin 2 → Type u) [∀ i, TopologicalSpace (X i)] : (∀ i, X i) ≃ₜ X 0 × X 1 where
toEquiv := piFinTwoEquiv X
continuous_toFun := (continuous_apply 0).prodMk (continuous_apply 1)
continuous_invFun := continuous_pi <| Fin.forall_fin_two.2 ⟨continuous_fst, continuous_snd⟩ | def | Topology | [
"Mathlib.Logic.Equiv.Fin.Basic",
"Mathlib.Topology.Connected.LocallyConnected",
"Mathlib.Topology.DenseEmbedding",
"Mathlib.Topology.Connected.TotallyDisconnected"
] | Mathlib/Topology/Homeomorph/Lemmas.lean | piFinTwo. | Homeomorphism between dependent functions `Π i : Fin 2, X i` and `X 0 × X 1`. |
@[simps! -fullyApplied]
finTwoArrow : (Fin 2 → X) ≃ₜ X × X :=
{ piFinTwo fun _ => X with toEquiv := finTwoArrowEquiv X } | def | Topology | [
"Mathlib.Logic.Equiv.Fin.Basic",
"Mathlib.Topology.Connected.LocallyConnected",
"Mathlib.Topology.DenseEmbedding",
"Mathlib.Topology.Connected.TotallyDisconnected"
] | Mathlib/Topology/Homeomorph/Lemmas.lean | finTwoArrow | Homeomorphism between `X² = Fin 2 → X` and `X × X`. |
@[simps!]
image (e : X ≃ₜ Y) (s : Set X) : s ≃ₜ e '' s where
continuous_toFun := e.continuous.continuousOn.mapsToRestrict (mapsTo_image _ _)
continuous_invFun := (e.symm.continuous.comp continuous_subtype_val).codRestrict _
toEquiv := e.toEquiv.image s | def | Topology | [
"Mathlib.Logic.Equiv.Fin.Basic",
"Mathlib.Topology.Connected.LocallyConnected",
"Mathlib.Topology.DenseEmbedding",
"Mathlib.Topology.Connected.TotallyDisconnected"
] | Mathlib/Topology/Homeomorph/Lemmas.lean | image | A subset of a topological space is homeomorphic to its image under a homeomorphism. |
@[simps! -fullyApplied]
Set.univ (X : Type*) [TopologicalSpace X] : (univ : Set X) ≃ₜ X where
toEquiv := Equiv.Set.univ X
continuous_toFun := continuous_subtype_val
continuous_invFun := continuous_id.subtype_mk _ | def | Topology | [
"Mathlib.Logic.Equiv.Fin.Basic",
"Mathlib.Topology.Connected.LocallyConnected",
"Mathlib.Topology.DenseEmbedding",
"Mathlib.Topology.Connected.TotallyDisconnected"
] | Mathlib/Topology/Homeomorph/Lemmas.lean | Set.univ | `Set.univ X` is homeomorphic to `X`. |
@[simps!]
Set.prod (s : Set X) (t : Set Y) : ↥(s ×ˢ t) ≃ₜ s × t where
toEquiv := Equiv.Set.prod s t
continuous_toFun :=
(continuous_subtype_val.fst.subtype_mk _).prodMk (continuous_subtype_val.snd.subtype_mk _)
continuous_invFun :=
(continuous_subtype_val.fst'.prodMk continuous_subtype_val.snd').subtype_mk _ | def | Topology | [
"Mathlib.Logic.Equiv.Fin.Basic",
"Mathlib.Topology.Connected.LocallyConnected",
"Mathlib.Topology.DenseEmbedding",
"Mathlib.Topology.Connected.TotallyDisconnected"
] | Mathlib/Topology/Homeomorph/Lemmas.lean | Set.prod | `s ×ˢ t` is homeomorphic to `s × t`. |
@[simps!]
piEquivPiSubtypeProd (p : ι → Prop) (Y : ι → Type*) [∀ i, TopologicalSpace (Y i)]
[DecidablePred p] : (∀ i, Y i) ≃ₜ (∀ i : { x // p x }, Y i) × ∀ i : { x // ¬p x }, Y i where
toEquiv := Equiv.piEquivPiSubtypeProd p Y
continuous_toFun := by
apply Continuous.prodMk <;> exact continuous_pi fun j => continuous_apply j.1
continuous_invFun :=
continuous_pi fun j => by
dsimp only [Equiv.piEquivPiSubtypeProd]; split_ifs
exacts [(continuous_apply _).comp continuous_fst, (continuous_apply _).comp continuous_snd]
variable [DecidableEq ι] (i : ι) | def | Topology | [
"Mathlib.Logic.Equiv.Fin.Basic",
"Mathlib.Topology.Connected.LocallyConnected",
"Mathlib.Topology.DenseEmbedding",
"Mathlib.Topology.Connected.TotallyDisconnected"
] | Mathlib/Topology/Homeomorph/Lemmas.lean | piEquivPiSubtypeProd | The topological space `Π i, Y i` can be split as a product by separating the indices in ι
depending on whether they satisfy a predicate p or not. |
@[simps!]
piSplitAt (Y : ι → Type*) [∀ j, TopologicalSpace (Y j)] :
(∀ j, Y j) ≃ₜ Y i × ∀ j : { j // j ≠ i }, Y j where
toEquiv := Equiv.piSplitAt i Y
continuous_toFun := (continuous_apply i).prodMk (continuous_pi fun j => continuous_apply j.1)
continuous_invFun :=
continuous_pi fun j => by
dsimp only [Equiv.piSplitAt]
split_ifs with h
· subst h
exact continuous_fst
· exact (continuous_apply _).comp continuous_snd
variable (Y) | def | Topology | [
"Mathlib.Logic.Equiv.Fin.Basic",
"Mathlib.Topology.Connected.LocallyConnected",
"Mathlib.Topology.DenseEmbedding",
"Mathlib.Topology.Connected.TotallyDisconnected"
] | Mathlib/Topology/Homeomorph/Lemmas.lean | piSplitAt | A product of topological spaces can be split as the binary product of one of the spaces and
the product of all the remaining spaces. |
@[simps!]
funSplitAt : (ι → Y) ≃ₜ Y × ({ j // j ≠ i } → Y) :=
piSplitAt i _ | def | Topology | [
"Mathlib.Logic.Equiv.Fin.Basic",
"Mathlib.Topology.Connected.LocallyConnected",
"Mathlib.Topology.DenseEmbedding",
"Mathlib.Topology.Connected.TotallyDisconnected"
] | Mathlib/Topology/Homeomorph/Lemmas.lean | funSplitAt | A product of copies of a topological space can be split as the binary product of one copy and
the product of all the remaining copies. |
@[simps! apply_coe]
noncomputable toHomeomorph {f : X → Y} (hf : IsEmbedding f) :
X ≃ₜ Set.range f :=
Equiv.ofInjective f hf.injective |>.toHomeomorphOfIsInducing <|
IsInducing.subtypeVal.of_comp_iff.mp hf.toIsInducing
@[deprecated (since := "2025-04-16")]
alias _root_.Homeomorph.ofIsEmbedding := toHomeomorph | def | Topology | [
"Mathlib.Logic.Equiv.Fin.Basic",
"Mathlib.Topology.Connected.LocallyConnected",
"Mathlib.Topology.DenseEmbedding",
"Mathlib.Topology.Connected.TotallyDisconnected"
] | Mathlib/Topology/Homeomorph/Lemmas.lean | toHomeomorph | Homeomorphism given an embedding. |
@[simps! apply]
noncomputable toHomeomorphOfSurjective {f : X → Y}
(hf : IsEmbedding f) (hsurj : Function.Surjective f) : X ≃ₜ Y :=
Equiv.ofBijective f ⟨hf.injective, hsurj⟩ |>.toHomeomorphOfIsInducing hf.toIsInducing
@[deprecated (since := "2025-04-16")]
alias toHomeomorph_of_surjective := toHomeomorphOfSurjective | def | Topology | [
"Mathlib.Logic.Equiv.Fin.Basic",
"Mathlib.Topology.Connected.LocallyConnected",
"Mathlib.Topology.DenseEmbedding",
"Mathlib.Topology.Connected.TotallyDisconnected"
] | Mathlib/Topology/Homeomorph/Lemmas.lean | toHomeomorphOfSurjective | A surjective embedding is a homeomorphism. |
noncomputable homeomorphImage {f : X → Y} (hf : IsEmbedding f) (s : Set X) : s ≃ₜ f '' s :=
(hf.comp .subtypeVal).toHomeomorph.trans <| .setCongr <| by simp [Set.range_comp] | def | Topology | [
"Mathlib.Logic.Equiv.Fin.Basic",
"Mathlib.Topology.Connected.LocallyConnected",
"Mathlib.Topology.DenseEmbedding",
"Mathlib.Topology.Connected.TotallyDisconnected"
] | Mathlib/Topology/Homeomorph/Lemmas.lean | homeomorphImage | A set is homeomorphic to its image under any embedding. |
continuous_symm_of_equiv_compact_to_t2 [CompactSpace X] [T2Space Y] {f : X ≃ Y}
(hf : Continuous f) : Continuous f.symm := by
rw [continuous_iff_isClosed]
intro C hC
have hC' : IsClosed (f '' C) := (hC.isCompact.image hf).isClosed
rwa [Equiv.image_eq_preimage] at hC' | theorem | Topology | [
"Mathlib.Logic.Equiv.Fin.Basic",
"Mathlib.Topology.Connected.LocallyConnected",
"Mathlib.Topology.DenseEmbedding",
"Mathlib.Topology.Connected.TotallyDisconnected"
] | Mathlib/Topology/Homeomorph/Lemmas.lean | continuous_symm_of_equiv_compact_to_t2 | null |
@[simps toEquiv]
homeoOfEquivCompactToT2 [CompactSpace X] [T2Space Y] {f : X ≃ Y} (hf : Continuous f) : X ≃ₜ Y :=
{ f with
continuous_toFun := hf
continuous_invFun := hf.continuous_symm_of_equiv_compact_to_t2 } | def | Topology | [
"Mathlib.Logic.Equiv.Fin.Basic",
"Mathlib.Topology.Connected.LocallyConnected",
"Mathlib.Topology.DenseEmbedding",
"Mathlib.Topology.Connected.TotallyDisconnected"
] | Mathlib/Topology/Homeomorph/Lemmas.lean | homeoOfEquivCompactToT2 | Continuous equivalences from a compact space to a T2 space are homeomorphisms.
This is not true when T2 is weakened to T1
(see `Continuous.homeoOfEquivCompactToT2.t1_counterexample`). |
@[simps! toEquiv apply symm_apply]
noncomputable homeomorph : X ≃ₜ Y where
continuous_toFun := hf.1
continuous_invFun := by
rw [← continuousOn_univ, ← hf.bijective.2.range_eq]
exact hf.isOpenMap.continuousOn_range_of_leftInverse (leftInverse_surjInv hf.bijective)
toEquiv := Equiv.ofBijective f hf.bijective | def | Topology | [
"Mathlib.Logic.Equiv.Fin.Basic",
"Mathlib.Topology.Connected.LocallyConnected",
"Mathlib.Topology.DenseEmbedding",
"Mathlib.Topology.Connected.TotallyDisconnected"
] | Mathlib/Topology/Homeomorph/Lemmas.lean | homeomorph | Bundled homeomorphism constructed from a map that is a homeomorphism. |
protected isClosedMap : IsClosedMap f := (hf.homeomorph f).isClosedMap | lemma | Topology | [
"Mathlib.Logic.Equiv.Fin.Basic",
"Mathlib.Topology.Connected.LocallyConnected",
"Mathlib.Topology.DenseEmbedding",
"Mathlib.Topology.Connected.TotallyDisconnected"
] | Mathlib/Topology/Homeomorph/Lemmas.lean | isClosedMap | null |
isInducing : IsInducing f := (hf.homeomorph f).isInducing | lemma | Topology | [
"Mathlib.Logic.Equiv.Fin.Basic",
"Mathlib.Topology.Connected.LocallyConnected",
"Mathlib.Topology.DenseEmbedding",
"Mathlib.Topology.Connected.TotallyDisconnected"
] | Mathlib/Topology/Homeomorph/Lemmas.lean | isInducing | null |
isQuotientMap : IsQuotientMap f := (hf.homeomorph f).isQuotientMap | lemma | Topology | [
"Mathlib.Logic.Equiv.Fin.Basic",
"Mathlib.Topology.Connected.LocallyConnected",
"Mathlib.Topology.DenseEmbedding",
"Mathlib.Topology.Connected.TotallyDisconnected"
] | Mathlib/Topology/Homeomorph/Lemmas.lean | isQuotientMap | null |
isEmbedding : IsEmbedding f := (hf.homeomorph f).isEmbedding | lemma | Topology | [
"Mathlib.Logic.Equiv.Fin.Basic",
"Mathlib.Topology.Connected.LocallyConnected",
"Mathlib.Topology.DenseEmbedding",
"Mathlib.Topology.Connected.TotallyDisconnected"
] | Mathlib/Topology/Homeomorph/Lemmas.lean | isEmbedding | null |
isOpenEmbedding : IsOpenEmbedding f := (hf.homeomorph f).isOpenEmbedding | lemma | Topology | [
"Mathlib.Logic.Equiv.Fin.Basic",
"Mathlib.Topology.Connected.LocallyConnected",
"Mathlib.Topology.DenseEmbedding",
"Mathlib.Topology.Connected.TotallyDisconnected"
] | Mathlib/Topology/Homeomorph/Lemmas.lean | isOpenEmbedding | null |
isClosedEmbedding : IsClosedEmbedding f := (hf.homeomorph f).isClosedEmbedding | lemma | Topology | [
"Mathlib.Logic.Equiv.Fin.Basic",
"Mathlib.Topology.Connected.LocallyConnected",
"Mathlib.Topology.DenseEmbedding",
"Mathlib.Topology.Connected.TotallyDisconnected"
] | Mathlib/Topology/Homeomorph/Lemmas.lean | isClosedEmbedding | null |
isDenseEmbedding : IsDenseEmbedding f := (hf.homeomorph f).isDenseEmbedding | lemma | Topology | [
"Mathlib.Logic.Equiv.Fin.Basic",
"Mathlib.Topology.Connected.LocallyConnected",
"Mathlib.Topology.DenseEmbedding",
"Mathlib.Topology.Connected.TotallyDisconnected"
] | Mathlib/Topology/Homeomorph/Lemmas.lean | isDenseEmbedding | null |
isHomeomorph_iff_exists_homeomorph : IsHomeomorph f ↔ ∃ h : X ≃ₜ Y, h = f :=
⟨fun hf => ⟨hf.homeomorph f, rfl⟩, fun ⟨h, h'⟩ => h' ▸ h.isHomeomorph⟩ | lemma | Topology | [
"Mathlib.Logic.Equiv.Fin.Basic",
"Mathlib.Topology.Connected.LocallyConnected",
"Mathlib.Topology.DenseEmbedding",
"Mathlib.Topology.Connected.TotallyDisconnected"
] | Mathlib/Topology/Homeomorph/Lemmas.lean | isHomeomorph_iff_exists_homeomorph | A map is a homeomorphism iff it is the map underlying a bundled homeomorphism `h : X ≃ₜ Y`. |
isHomeomorph_iff_exists_inverse : IsHomeomorph f ↔ Continuous f ∧ ∃ g : Y → X,
LeftInverse g f ∧ RightInverse g f ∧ Continuous g := by
refine ⟨fun hf ↦ ⟨hf.continuous, ?_⟩, fun ⟨hf, g, hg⟩ ↦ ?_⟩
· let h := hf.homeomorph f
exact ⟨h.symm, h.left_inv, h.right_inv, h.continuous_invFun⟩
· exact (Homeomorph.mk ⟨f, g, hg.1, hg.2.1⟩ hf hg.2.2).isHomeomorph | lemma | Topology | [
"Mathlib.Logic.Equiv.Fin.Basic",
"Mathlib.Topology.Connected.LocallyConnected",
"Mathlib.Topology.DenseEmbedding",
"Mathlib.Topology.Connected.TotallyDisconnected"
] | Mathlib/Topology/Homeomorph/Lemmas.lean | isHomeomorph_iff_exists_inverse | A map is a homeomorphism iff it is continuous and has a continuous inverse. |
isHomeomorph_iff_isEmbedding_surjective : IsHomeomorph f ↔ IsEmbedding f ∧ Surjective f where
mp hf := ⟨hf.isEmbedding, hf.surjective⟩
mpr h := ⟨h.1.continuous, ((isOpenEmbedding_iff f).2 ⟨h.1, h.2.range_eq ▸ isOpen_univ⟩).isOpenMap,
h.1.injective, h.2⟩ | lemma | Topology | [
"Mathlib.Logic.Equiv.Fin.Basic",
"Mathlib.Topology.Connected.LocallyConnected",
"Mathlib.Topology.DenseEmbedding",
"Mathlib.Topology.Connected.TotallyDisconnected"
] | Mathlib/Topology/Homeomorph/Lemmas.lean | isHomeomorph_iff_isEmbedding_surjective | A map is a homeomorphism iff it is a surjective embedding. |
isHomeomorph_iff_continuous_isClosedMap_bijective : IsHomeomorph f ↔
Continuous f ∧ IsClosedMap f ∧ Function.Bijective f :=
⟨fun hf => ⟨hf.continuous, hf.isClosedMap, hf.bijective⟩, fun ⟨hf, hf', hf''⟩ =>
⟨hf, fun _ hu => isClosed_compl_iff.1 (image_compl_eq hf'' ▸ hf' _ hu.isClosed_compl), hf''⟩⟩ | lemma | Topology | [
"Mathlib.Logic.Equiv.Fin.Basic",
"Mathlib.Topology.Connected.LocallyConnected",
"Mathlib.Topology.DenseEmbedding",
"Mathlib.Topology.Connected.TotallyDisconnected"
] | Mathlib/Topology/Homeomorph/Lemmas.lean | isHomeomorph_iff_continuous_isClosedMap_bijective | A map is a homeomorphism iff it is continuous, closed and bijective. |
isHomeomorph_iff_continuous_bijective [CompactSpace X] [T2Space Y] :
IsHomeomorph f ↔ Continuous f ∧ Bijective f := by
rw [isHomeomorph_iff_continuous_isClosedMap_bijective]
refine and_congr_right fun hf ↦ ?_
rw [eq_true hf.isClosedMap, true_and] | lemma | Topology | [
"Mathlib.Logic.Equiv.Fin.Basic",
"Mathlib.Topology.Connected.LocallyConnected",
"Mathlib.Topology.DenseEmbedding",
"Mathlib.Topology.Connected.TotallyDisconnected"
] | Mathlib/Topology/Homeomorph/Lemmas.lean | isHomeomorph_iff_continuous_bijective | A map from a compact space to a T2 space is a homeomorphism iff it is continuous and
bijective. |
IsHomeomorph.sumMap {g : Z → W} (hf : IsHomeomorph f) (hg : IsHomeomorph g) :
IsHomeomorph (Sum.map f g) := ⟨hf.1.sumMap hg.1, hf.2.sumMap hg.2, hf.3.sumMap hg.3⟩ | lemma | Topology | [
"Mathlib.Logic.Equiv.Fin.Basic",
"Mathlib.Topology.Connected.LocallyConnected",
"Mathlib.Topology.DenseEmbedding",
"Mathlib.Topology.Connected.TotallyDisconnected"
] | Mathlib/Topology/Homeomorph/Lemmas.lean | IsHomeomorph.sumMap | null |
IsHomeomorph.prodMap {g : Z → W} (hf : IsHomeomorph f) (hg : IsHomeomorph g) :
IsHomeomorph (Prod.map f g) := ⟨hf.1.prodMap hg.1, hf.2.prodMap hg.2, hf.3.prodMap hg.3⟩ | lemma | Topology | [
"Mathlib.Logic.Equiv.Fin.Basic",
"Mathlib.Topology.Connected.LocallyConnected",
"Mathlib.Topology.DenseEmbedding",
"Mathlib.Topology.Connected.TotallyDisconnected"
] | Mathlib/Topology/Homeomorph/Lemmas.lean | IsHomeomorph.prodMap | null |
IsHomeomorph.sigmaMap {ι κ : Type*} {X : ι → Type*} {Y : κ → Type*}
[∀ i, TopologicalSpace (X i)] [∀ i, TopologicalSpace (Y i)] {f : ι → κ}
(hf : Bijective f) {g : (i : ι) → X i → Y (f i)} (hg : ∀ i, IsHomeomorph (g i)) :
IsHomeomorph (Sigma.map f g) := by
simp_rw [isHomeomorph_iff_isEmbedding_surjective,] at hg ⊢
exact ⟨(isEmbedding_sigmaMap hf.1).2 fun i ↦ (hg i).1, hf.2.sigma_map fun i ↦ (hg i).2⟩ | lemma | Topology | [
"Mathlib.Logic.Equiv.Fin.Basic",
"Mathlib.Topology.Connected.LocallyConnected",
"Mathlib.Topology.DenseEmbedding",
"Mathlib.Topology.Connected.TotallyDisconnected"
] | Mathlib/Topology/Homeomorph/Lemmas.lean | IsHomeomorph.sigmaMap | null |
IsHomeomorph.pi_map {ι : Type*} {X Y : ι → Type*} [∀ i, TopologicalSpace (X i)]
[∀ i, TopologicalSpace (Y i)] {f : (i : ι) → X i → Y i} (h : ∀ i, IsHomeomorph (f i)) :
IsHomeomorph (fun (x : ∀ i, X i) i ↦ f i (x i)) :=
(Homeomorph.piCongrRight fun i ↦ (h i).homeomorph (f i)).isHomeomorph | lemma | Topology | [
"Mathlib.Logic.Equiv.Fin.Basic",
"Mathlib.Topology.Connected.LocallyConnected",
"Mathlib.Topology.DenseEmbedding",
"Mathlib.Topology.Connected.TotallyDisconnected"
] | Mathlib/Topology/Homeomorph/Lemmas.lean | IsHomeomorph.pi_map | null |
Homotopy (f₀ f₁ : C(X, Y)) extends C(I × X, Y) where
/-- value of the homotopy at 0 -/
map_zero_left : ∀ x, toFun (0, x) = f₀ x
/-- value of the homotopy at 1 -/
map_one_left : ∀ x, toFun (1, x) = f₁ x | structure | Topology | [
"Mathlib.Topology.Order.ProjIcc",
"Mathlib.Topology.ContinuousMap.Ordered",
"Mathlib.Topology.CompactOpen",
"Mathlib.Topology.UnitInterval"
] | Mathlib/Topology/Homotopy/Basic.lean | Homotopy | `ContinuousMap.Homotopy f₀ f₁` is the type of homotopies from `f₀` to `f₁`.
When possible, instead of parametrizing results over `(f : Homotopy f₀ f₁)`,
you should parametrize over `{F : Type*} [HomotopyLike F f₀ f₁] (f : F)`.
When you extend this structure, make sure to extend `ContinuousMap.HomotopyLike`. |
HomotopyLike {X Y : outParam Type*} [TopologicalSpace X] [TopologicalSpace Y]
(F : Type*) (f₀ f₁ : outParam <| C(X, Y)) [FunLike F (I × X) Y] : Prop
extends ContinuousMapClass F (I × X) Y where
/-- value of the homotopy at 0 -/
map_zero_left (f : F) : ∀ x, f (0, x) = f₀ x
/-- value of the homotopy at 1 -/
map_one_left (f : F) : ∀ x, f (1, x) = f₁ x | class | Topology | [
"Mathlib.Topology.Order.ProjIcc",
"Mathlib.Topology.ContinuousMap.Ordered",
"Mathlib.Topology.CompactOpen",
"Mathlib.Topology.UnitInterval"
] | Mathlib/Topology/Homotopy/Basic.lean | HomotopyLike | `ContinuousMap.HomotopyLike F f₀ f₁` states that `F` is a type of homotopies between `f₀` and
`f₁`.
You should extend this class when you extend `ContinuousMap.Homotopy`. |
instFunLike : FunLike (Homotopy f₀ f₁) (I × X) Y where
coe f := f.toFun
coe_injective' f g h := by
obtain ⟨⟨_, _⟩, _⟩ := f
obtain ⟨⟨_, _⟩, _⟩ := g
congr | instance | Topology | [
"Mathlib.Topology.Order.ProjIcc",
"Mathlib.Topology.ContinuousMap.Ordered",
"Mathlib.Topology.CompactOpen",
"Mathlib.Topology.UnitInterval"
] | Mathlib/Topology/Homotopy/Basic.lean | instFunLike | null |
@[ext]
ext {F G : Homotopy f₀ f₁} (h : ∀ x, F x = G x) : F = G :=
DFunLike.ext _ _ h | theorem | Topology | [
"Mathlib.Topology.Order.ProjIcc",
"Mathlib.Topology.ContinuousMap.Ordered",
"Mathlib.Topology.CompactOpen",
"Mathlib.Topology.UnitInterval"
] | Mathlib/Topology/Homotopy/Basic.lean | ext | null |
Simps.apply (F : Homotopy f₀ f₁) : I × X → Y :=
F
initialize_simps_projections Homotopy (toFun → apply, -toContinuousMap) | def | Topology | [
"Mathlib.Topology.Order.ProjIcc",
"Mathlib.Topology.ContinuousMap.Ordered",
"Mathlib.Topology.CompactOpen",
"Mathlib.Topology.UnitInterval"
] | Mathlib/Topology/Homotopy/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 continuous (F : Homotopy f₀ f₁) : Continuous F :=
F.continuous_toFun
@[simp] | theorem | Topology | [
"Mathlib.Topology.Order.ProjIcc",
"Mathlib.Topology.ContinuousMap.Ordered",
"Mathlib.Topology.CompactOpen",
"Mathlib.Topology.UnitInterval"
] | Mathlib/Topology/Homotopy/Basic.lean | continuous | Deprecated. Use `map_continuous` instead. |
apply_zero (F : Homotopy f₀ f₁) (x : X) : F (0, x) = f₀ x :=
F.map_zero_left x
@[simp] | theorem | Topology | [
"Mathlib.Topology.Order.ProjIcc",
"Mathlib.Topology.ContinuousMap.Ordered",
"Mathlib.Topology.CompactOpen",
"Mathlib.Topology.UnitInterval"
] | Mathlib/Topology/Homotopy/Basic.lean | apply_zero | null |
apply_one (F : Homotopy f₀ f₁) (x : X) : F (1, x) = f₁ x :=
F.map_one_left x
@[simp] | theorem | Topology | [
"Mathlib.Topology.Order.ProjIcc",
"Mathlib.Topology.ContinuousMap.Ordered",
"Mathlib.Topology.CompactOpen",
"Mathlib.Topology.UnitInterval"
] | Mathlib/Topology/Homotopy/Basic.lean | apply_one | null |
coe_toContinuousMap (F : Homotopy f₀ f₁) : ⇑F.toContinuousMap = F :=
rfl | theorem | Topology | [
"Mathlib.Topology.Order.ProjIcc",
"Mathlib.Topology.ContinuousMap.Ordered",
"Mathlib.Topology.CompactOpen",
"Mathlib.Topology.UnitInterval"
] | Mathlib/Topology/Homotopy/Basic.lean | coe_toContinuousMap | null |
curry (F : Homotopy f₀ f₁) : C(I, C(X, Y)) :=
F.toContinuousMap.curry
@[simp] | def | Topology | [
"Mathlib.Topology.Order.ProjIcc",
"Mathlib.Topology.ContinuousMap.Ordered",
"Mathlib.Topology.CompactOpen",
"Mathlib.Topology.UnitInterval"
] | Mathlib/Topology/Homotopy/Basic.lean | curry | Currying a homotopy to a continuous function from `I` to `C(X, Y)`. |
curry_apply (F : Homotopy f₀ f₁) (t : I) (x : X) : F.curry t x = F (t, x) :=
rfl
@[simp] theorem curry_zero (F : Homotopy f₀ f₁) : F.curry 0 = f₀ := by ext; simp
@[simp] theorem curry_one (F : Homotopy f₀ f₁) : F.curry 1 = f₁ := by ext; simp | theorem | Topology | [
"Mathlib.Topology.Order.ProjIcc",
"Mathlib.Topology.ContinuousMap.Ordered",
"Mathlib.Topology.CompactOpen",
"Mathlib.Topology.UnitInterval"
] | Mathlib/Topology/Homotopy/Basic.lean | curry_apply | null |
extend (F : Homotopy f₀ f₁) : C(ℝ, C(X, Y)) :=
F.curry.IccExtend zero_le_one | def | Topology | [
"Mathlib.Topology.Order.ProjIcc",
"Mathlib.Topology.ContinuousMap.Ordered",
"Mathlib.Topology.CompactOpen",
"Mathlib.Topology.UnitInterval"
] | Mathlib/Topology/Homotopy/Basic.lean | extend | Continuously extending a curried homotopy to a function from `ℝ` to `C(X, Y)`. |
extend_apply_of_le_zero (F : Homotopy f₀ f₁) {t : ℝ} (ht : t ≤ 0) (x : X) :
F.extend t x = f₀ x := by
rw [← F.apply_zero]
exact ContinuousMap.congr_fun (Set.IccExtend_of_le_left (zero_le_one' ℝ) F.curry ht) x | theorem | Topology | [
"Mathlib.Topology.Order.ProjIcc",
"Mathlib.Topology.ContinuousMap.Ordered",
"Mathlib.Topology.CompactOpen",
"Mathlib.Topology.UnitInterval"
] | Mathlib/Topology/Homotopy/Basic.lean | extend_apply_of_le_zero | null |
extend_apply_of_one_le (F : Homotopy f₀ f₁) {t : ℝ} (ht : 1 ≤ t) (x : X) :
F.extend t x = f₁ x := by
rw [← F.apply_one]
exact ContinuousMap.congr_fun (Set.IccExtend_of_right_le (zero_le_one' ℝ) F.curry ht) x | theorem | Topology | [
"Mathlib.Topology.Order.ProjIcc",
"Mathlib.Topology.ContinuousMap.Ordered",
"Mathlib.Topology.CompactOpen",
"Mathlib.Topology.UnitInterval"
] | Mathlib/Topology/Homotopy/Basic.lean | extend_apply_of_one_le | null |
extend_apply_coe (F : Homotopy f₀ f₁) (t : I) (x : X) : F.extend t x = F (t, x) :=
ContinuousMap.congr_fun (Set.IccExtend_val (zero_le_one' ℝ) F.curry t) x
@[simp] | theorem | Topology | [
"Mathlib.Topology.Order.ProjIcc",
"Mathlib.Topology.ContinuousMap.Ordered",
"Mathlib.Topology.CompactOpen",
"Mathlib.Topology.UnitInterval"
] | Mathlib/Topology/Homotopy/Basic.lean | extend_apply_coe | null |
extend_of_mem_I (F : Homotopy f₀ f₁) {t : ℝ} (ht : t ∈ I) :
F.extend t = F.curry ⟨t, ht⟩ :=
Set.IccExtend_of_mem (zero_le_one' ℝ) F.curry ht | theorem | Topology | [
"Mathlib.Topology.Order.ProjIcc",
"Mathlib.Topology.ContinuousMap.Ordered",
"Mathlib.Topology.CompactOpen",
"Mathlib.Topology.UnitInterval"
] | Mathlib/Topology/Homotopy/Basic.lean | extend_of_mem_I | null |
extend_zero (F : Homotopy f₀ f₁) : F.extend 0 = f₀ := by simp | theorem | Topology | [
"Mathlib.Topology.Order.ProjIcc",
"Mathlib.Topology.ContinuousMap.Ordered",
"Mathlib.Topology.CompactOpen",
"Mathlib.Topology.UnitInterval"
] | Mathlib/Topology/Homotopy/Basic.lean | extend_zero | null |
extend_one (F : Homotopy f₀ f₁) : F.extend 1 = f₁ := by simp | theorem | Topology | [
"Mathlib.Topology.Order.ProjIcc",
"Mathlib.Topology.ContinuousMap.Ordered",
"Mathlib.Topology.CompactOpen",
"Mathlib.Topology.UnitInterval"
] | Mathlib/Topology/Homotopy/Basic.lean | extend_one | null |
extend_apply_of_mem_I (F : Homotopy f₀ f₁) {t : ℝ} (ht : t ∈ I) (x : X) :
F.extend t x = F (⟨t, ht⟩, x) := by
simp [ht] | theorem | Topology | [
"Mathlib.Topology.Order.ProjIcc",
"Mathlib.Topology.ContinuousMap.Ordered",
"Mathlib.Topology.CompactOpen",
"Mathlib.Topology.UnitInterval"
] | Mathlib/Topology/Homotopy/Basic.lean | extend_apply_of_mem_I | null |
protected congr_fun {F G : Homotopy f₀ f₁} (h : F = G) (x : I × X) : F x = G x :=
ContinuousMap.congr_fun (congr_arg _ h) x | theorem | Topology | [
"Mathlib.Topology.Order.ProjIcc",
"Mathlib.Topology.ContinuousMap.Ordered",
"Mathlib.Topology.CompactOpen",
"Mathlib.Topology.UnitInterval"
] | Mathlib/Topology/Homotopy/Basic.lean | congr_fun | null |
protected congr_arg (F : Homotopy f₀ f₁) {x y : I × X} (h : x = y) : F x = F y :=
F.toContinuousMap.congr_arg h | theorem | Topology | [
"Mathlib.Topology.Order.ProjIcc",
"Mathlib.Topology.ContinuousMap.Ordered",
"Mathlib.Topology.CompactOpen",
"Mathlib.Topology.UnitInterval"
] | Mathlib/Topology/Homotopy/Basic.lean | congr_arg | null |
@[simps]
refl (f : C(X, Y)) : Homotopy f f where
toFun x := f x.2
map_zero_left _ := rfl
map_one_left _ := rfl | def | Topology | [
"Mathlib.Topology.Order.ProjIcc",
"Mathlib.Topology.ContinuousMap.Ordered",
"Mathlib.Topology.CompactOpen",
"Mathlib.Topology.UnitInterval"
] | Mathlib/Topology/Homotopy/Basic.lean | refl | Given a continuous function `f`, we can define a `Homotopy f f` by `F (t, x) = f x` |
@[simps]
symm {f₀ f₁ : C(X, Y)} (F : Homotopy f₀ f₁) : Homotopy f₁ f₀ where
toFun x := F (σ x.1, x.2)
map_zero_left := by simp
map_one_left := by norm_num
@[simp] | def | Topology | [
"Mathlib.Topology.Order.ProjIcc",
"Mathlib.Topology.ContinuousMap.Ordered",
"Mathlib.Topology.CompactOpen",
"Mathlib.Topology.UnitInterval"
] | Mathlib/Topology/Homotopy/Basic.lean | symm | Given a `Homotopy f₀ f₁`, we can define a `Homotopy f₁ f₀` by reversing the homotopy. |
symm_symm {f₀ f₁ : C(X, Y)} (F : Homotopy f₀ f₁) : F.symm.symm = F := by
ext
simp | theorem | Topology | [
"Mathlib.Topology.Order.ProjIcc",
"Mathlib.Topology.ContinuousMap.Ordered",
"Mathlib.Topology.CompactOpen",
"Mathlib.Topology.UnitInterval"
] | Mathlib/Topology/Homotopy/Basic.lean | symm_symm | null |
symm_bijective {f₀ f₁ : C(X, Y)} :
Function.Bijective (Homotopy.symm : Homotopy f₀ f₁ → Homotopy f₁ f₀) :=
Function.bijective_iff_has_inverse.mpr ⟨_, symm_symm, symm_symm⟩ | theorem | Topology | [
"Mathlib.Topology.Order.ProjIcc",
"Mathlib.Topology.ContinuousMap.Ordered",
"Mathlib.Topology.CompactOpen",
"Mathlib.Topology.UnitInterval"
] | Mathlib/Topology/Homotopy/Basic.lean | symm_bijective | null |
trans {f₀ f₁ f₂ : C(X, Y)} (F : Homotopy f₀ f₁) (G : Homotopy f₁ f₂) : Homotopy f₀ f₂ where
toFun x := if (x.1 : ℝ) ≤ 1 / 2 then F.extend (2 * x.1) x.2 else G.extend (2 * x.1 - 1) x.2
continuous_toFun := by
refine
continuous_if_le (by fun_prop) continuous_const
(F.continuous.comp (by continuity)).continuousOn
(G.continuous.comp (by continuity)).continuousOn ?_
rintro x hx
norm_num [hx]
map_zero_left x := by norm_num
map_one_left x := by norm_num | def | Topology | [
"Mathlib.Topology.Order.ProjIcc",
"Mathlib.Topology.ContinuousMap.Ordered",
"Mathlib.Topology.CompactOpen",
"Mathlib.Topology.UnitInterval"
] | Mathlib/Topology/Homotopy/Basic.lean | trans | Given `Homotopy f₀ f₁` and `Homotopy f₁ f₂`, we can define a `Homotopy f₀ f₂` by putting the first
homotopy on `[0, 1/2]` and the second on `[1/2, 1]`. |
trans_apply {f₀ f₁ f₂ : C(X, Y)} (F : Homotopy f₀ f₁) (G : Homotopy f₁ f₂) (x : I × X) :
(F.trans G) x =
if h : (x.1 : ℝ) ≤ 1 / 2 then
F (⟨2 * x.1, (unitInterval.mul_pos_mem_iff zero_lt_two).2 ⟨x.1.2.1, h⟩⟩, x.2)
else
G (⟨2 * x.1 - 1, unitInterval.two_mul_sub_one_mem_iff.2 ⟨(not_le.1 h).le, x.1.2.2⟩⟩, x.2) :=
show ite _ _ _ = _ by
split_ifs <;>
· rw [extend, ContinuousMap.coe_IccExtend, Set.IccExtend_of_mem]
rfl | theorem | Topology | [
"Mathlib.Topology.Order.ProjIcc",
"Mathlib.Topology.ContinuousMap.Ordered",
"Mathlib.Topology.CompactOpen",
"Mathlib.Topology.UnitInterval"
] | Mathlib/Topology/Homotopy/Basic.lean | trans_apply | null |
symm_trans {f₀ f₁ f₂ : C(X, Y)} (F : Homotopy f₀ f₁) (G : Homotopy f₁ f₂) :
(F.trans G).symm = G.symm.trans F.symm := by
ext ⟨t, _⟩
rw [trans_apply, symm_apply, trans_apply]
simp only [coe_symm_eq, symm_apply]
split_ifs with h₁ h₂ h₂
· have ht : (t : ℝ) = 1 / 2 := by linarith
norm_num [ht]
· congr 2
apply Subtype.ext
simp only [coe_symm_eq]
linarith
· congr 2
apply Subtype.ext
simp only [coe_symm_eq]
linarith
· exfalso
linarith | theorem | Topology | [
"Mathlib.Topology.Order.ProjIcc",
"Mathlib.Topology.ContinuousMap.Ordered",
"Mathlib.Topology.CompactOpen",
"Mathlib.Topology.UnitInterval"
] | Mathlib/Topology/Homotopy/Basic.lean | symm_trans | null |
@[simps]
cast {f₀ f₁ g₀ g₁ : C(X, Y)} (F : Homotopy f₀ f₁) (h₀ : f₀ = g₀) (h₁ : f₁ = g₁) :
Homotopy g₀ g₁ where
toFun := F
map_zero_left := by simp [← h₀]
map_one_left := by simp [← h₁] | def | Topology | [
"Mathlib.Topology.Order.ProjIcc",
"Mathlib.Topology.ContinuousMap.Ordered",
"Mathlib.Topology.CompactOpen",
"Mathlib.Topology.UnitInterval"
] | Mathlib/Topology/Homotopy/Basic.lean | cast | Casting a `Homotopy f₀ f₁` to a `Homotopy g₀ g₁` where `f₀ = g₀` and `f₁ = g₁`. |
@[simps]
comp {f₀ f₁ : C(X, Y)} {g₀ g₁ : C(Y, Z)} (G : Homotopy g₀ g₁) (F : Homotopy f₀ f₁) :
Homotopy (g₀.comp f₀) (g₁.comp f₁) where
toFun x := G (x.1, F x)
map_zero_left := by simp
map_one_left := by simp | def | Topology | [
"Mathlib.Topology.Order.ProjIcc",
"Mathlib.Topology.ContinuousMap.Ordered",
"Mathlib.Topology.CompactOpen",
"Mathlib.Topology.UnitInterval"
] | Mathlib/Topology/Homotopy/Basic.lean | comp | If we have a `Homotopy g₀ g₁` and a `Homotopy f₀ f₁`, then we can compose them and get a
`Homotopy (g₀.comp f₀) (g₁.comp f₁)`. |
@[simps!]
compContinuousMap {g₀ g₁ : C(Y, Z)} (G : Homotopy g₀ g₁) (f : C(X, Y)) :
Homotopy (g₀.comp f) (g₁.comp f) :=
G.comp (.refl f) | def | Topology | [
"Mathlib.Topology.Order.ProjIcc",
"Mathlib.Topology.ContinuousMap.Ordered",
"Mathlib.Topology.CompactOpen",
"Mathlib.Topology.UnitInterval"
] | Mathlib/Topology/Homotopy/Basic.lean | compContinuousMap | Composition of a `Homotopy g₀ g₁` and `f : C(X, Y)` as a homotopy between `g₀.comp f` and
`g₁.comp f`. |
@[simps!, deprecated comp (since := "2025-05-12")]
hcomp {f₀ f₁ : C(X, Y)} {g₀ g₁ : C(Y, Z)} (F : Homotopy f₀ f₁) (G : Homotopy g₀ g₁) :
Homotopy (g₀.comp f₀) (g₁.comp f₁) :=
G.comp F | def | Topology | [
"Mathlib.Topology.Order.ProjIcc",
"Mathlib.Topology.ContinuousMap.Ordered",
"Mathlib.Topology.CompactOpen",
"Mathlib.Topology.UnitInterval"
] | Mathlib/Topology/Homotopy/Basic.lean | hcomp | If we have a `Homotopy f₀ f₁` and a `Homotopy g₀ g₁`, then we can compose them and get a
`Homotopy (g₀.comp f₀) (g₁.comp f₁)`. |
prodMap {f₀ f₁ : C(X, Y)} {g₀ g₁ : C(Z, Z')} (F : Homotopy f₀ f₁) (G : Homotopy g₀ g₁) :
Homotopy (f₀.prodMap g₀) (f₁.prodMap g₁) :=
.prodMk (F.compContinuousMap .fst) (G.compContinuousMap .snd) | def | Topology | [
"Mathlib.Topology.Order.ProjIcc",
"Mathlib.Topology.ContinuousMap.Ordered",
"Mathlib.Topology.CompactOpen",
"Mathlib.Topology.UnitInterval"
] | Mathlib/Topology/Homotopy/Basic.lean | prodMap | Let `F` be a homotopy between `f₀ : C(X, Y)` and `f₁ : C(X, Y)`. Let `G` be a homotopy between
`g₀ : C(X, Z)` and `g₁ : C(X, Z)`. Then `F.prodMk G` is the homotopy between `f₀.prodMk g₀` and
`f₁.prodMk g₁` that sends `p` to `(F p, G p)`. -/
nonrec def prodMk {f₀ f₁ : C(X, Y)} {g₀ g₁ : C(X, Z)} (F : Homotopy f₀ f₁) (G : Homotopy g₀ g₁) :
Homotopy (f₀.prodMk g₀) (f₁.prodMk g₁) where
toContinuousMap := F.prodMk G
map_zero_left _ := Prod.ext (F.map_zero_left _) (G.map_zero_left _)
map_one_left _ := Prod.ext (F.map_one_left _) (G.map_one_left _)
/-- Let `F` be a homotopy between `f₀ : C(X, Y)` and `f₁ : C(X, Y)`. Let `G` be a homotopy between
`g₀ : C(Z, Z')` and `g₁ : C(Z, Z')`. Then `F.prodMap G` is the homotopy between `f₀.prodMap g₀` and
`f₁.prodMap g₁` that sends `(t, x, z)` to `(F (t, x), G (t, z))`. |
protected pi {Y : ι → Type*} [∀ i, TopologicalSpace (Y i)] {f₀ f₁ : ∀ i, C(X, Y i)}
(F : ∀ i, Homotopy (f₀ i) (f₁ i)) :
Homotopy (.pi f₀) (.pi f₁) where
toContinuousMap := .pi fun i ↦ F i
map_zero_left x := funext fun i ↦ (F i).map_zero_left x
map_one_left x := funext fun i ↦ (F i).map_one_left x | def | Topology | [
"Mathlib.Topology.Order.ProjIcc",
"Mathlib.Topology.ContinuousMap.Ordered",
"Mathlib.Topology.CompactOpen",
"Mathlib.Topology.UnitInterval"
] | Mathlib/Topology/Homotopy/Basic.lean | pi | Given a family of homotopies `F i` between `f₀ i : C(X, Y i)` and `f₁ i : C(X, Y i)`, returns a
homotopy between `ContinuousMap.pi f₀` and `ContinuousMap.pi f₁`. |
protected piMap {X Y : ι → Type*} [∀ i, TopologicalSpace (X i)] [∀ i, TopologicalSpace (Y i)]
{f₀ f₁ : ∀ i, C(X i, Y i)} (F : ∀ i, Homotopy (f₀ i) (f₁ i)) :
Homotopy (.piMap f₀) (.piMap f₁) :=
.pi fun i ↦ (F i).compContinuousMap <| .eval i | def | Topology | [
"Mathlib.Topology.Order.ProjIcc",
"Mathlib.Topology.ContinuousMap.Ordered",
"Mathlib.Topology.CompactOpen",
"Mathlib.Topology.UnitInterval"
] | Mathlib/Topology/Homotopy/Basic.lean | piMap | Given a family of homotopies `F i` between `f₀ i : C(X i, Y i)` and `f₁ i : C(X i, Y i)`,
returns a homotopy between `ContinuousMap.piMap f₀` and `ContinuousMap.piMap f₁`. |
Homotopic (f₀ f₁ : C(X, Y)) : Prop :=
Nonempty (Homotopy f₀ f₁) | def | Topology | [
"Mathlib.Topology.Order.ProjIcc",
"Mathlib.Topology.ContinuousMap.Ordered",
"Mathlib.Topology.CompactOpen",
"Mathlib.Topology.UnitInterval"
] | Mathlib/Topology/Homotopy/Basic.lean | Homotopic | Given continuous maps `f₀` and `f₁`, we say `f₀` and `f₁` are homotopic if there exists a
`Homotopy f₀ f₁`. |
@[refl]
refl (f : C(X, Y)) : Homotopic f f :=
⟨Homotopy.refl f⟩
@[symm] | theorem | Topology | [
"Mathlib.Topology.Order.ProjIcc",
"Mathlib.Topology.ContinuousMap.Ordered",
"Mathlib.Topology.CompactOpen",
"Mathlib.Topology.UnitInterval"
] | Mathlib/Topology/Homotopy/Basic.lean | refl | null |
symm ⦃f g : C(X, Y)⦄ (h : Homotopic f g) : Homotopic g f :=
h.map Homotopy.symm
@[trans] | theorem | Topology | [
"Mathlib.Topology.Order.ProjIcc",
"Mathlib.Topology.ContinuousMap.Ordered",
"Mathlib.Topology.CompactOpen",
"Mathlib.Topology.UnitInterval"
] | Mathlib/Topology/Homotopy/Basic.lean | symm | null |
trans ⦃f g h : C(X, Y)⦄ (h₀ : Homotopic f g) (h₁ : Homotopic g h) : Homotopic f h :=
h₀.map2 Homotopy.trans h₁ | theorem | Topology | [
"Mathlib.Topology.Order.ProjIcc",
"Mathlib.Topology.ContinuousMap.Ordered",
"Mathlib.Topology.CompactOpen",
"Mathlib.Topology.UnitInterval"
] | Mathlib/Topology/Homotopy/Basic.lean | trans | null |
comp {g₀ g₁ : C(Y, Z)} {f₀ f₁ : C(X, Y)} (hg : Homotopic g₀ g₁) (hf : Homotopic f₀ f₁) :
Homotopic (g₀.comp f₀) (g₁.comp f₁) :=
hg.map2 Homotopy.comp hf
@[deprecated comp (since := "2025-05-12")] | theorem | Topology | [
"Mathlib.Topology.Order.ProjIcc",
"Mathlib.Topology.ContinuousMap.Ordered",
"Mathlib.Topology.CompactOpen",
"Mathlib.Topology.UnitInterval"
] | Mathlib/Topology/Homotopy/Basic.lean | comp | null |
hcomp {f₀ f₁ : C(X, Y)} {g₀ g₁ : C(Y, Z)} (h₀ : Homotopic f₀ f₁) (h₁ : Homotopic g₀ g₁) :
Homotopic (g₀.comp f₀) (g₁.comp f₁) :=
h₁.comp h₀ | theorem | Topology | [
"Mathlib.Topology.Order.ProjIcc",
"Mathlib.Topology.ContinuousMap.Ordered",
"Mathlib.Topology.CompactOpen",
"Mathlib.Topology.UnitInterval"
] | Mathlib/Topology/Homotopy/Basic.lean | hcomp | null |
equivalence : Equivalence (@Homotopic X Y _ _) :=
⟨refl, by apply symm, by apply trans⟩
nonrec theorem prodMk {f₀ f₁ : C(X, Y)} {g₀ g₁ : C(X, Z)} :
Homotopic f₀ f₁ → Homotopic g₀ g₁ → Homotopic (f₀.prodMk g₀) (f₁.prodMk g₁)
| ⟨F⟩, ⟨G⟩ => ⟨F.prodMk G⟩
nonrec theorem prodMap {f₀ f₁ : C(X, Y)} {g₀ g₁ : C(Z, Z')} :
Homotopic f₀ f₁ → Homotopic g₀ g₁ → Homotopic (f₀.prodMap g₀) (f₁.prodMap g₁)
| ⟨F⟩, ⟨G⟩ => ⟨F.prodMap G⟩ | theorem | Topology | [
"Mathlib.Topology.Order.ProjIcc",
"Mathlib.Topology.ContinuousMap.Ordered",
"Mathlib.Topology.CompactOpen",
"Mathlib.Topology.UnitInterval"
] | Mathlib/Topology/Homotopy/Basic.lean | equivalence | null |
protected pi {Y : ι → Type*} [∀ i, TopologicalSpace (Y i)] {f₀ f₁ : ∀ i, C(X, Y i)}
(F : ∀ i, Homotopic (f₀ i) (f₁ i)) :
Homotopic (.pi f₀) (.pi f₁) :=
⟨.pi fun i ↦ (F i).some⟩ | theorem | Topology | [
"Mathlib.Topology.Order.ProjIcc",
"Mathlib.Topology.ContinuousMap.Ordered",
"Mathlib.Topology.CompactOpen",
"Mathlib.Topology.UnitInterval"
] | Mathlib/Topology/Homotopy/Basic.lean | pi | If each `f₀ i : C(X, Y i)` is homotopic to `f₁ i : C(X, Y i)`, then `ContinuousMap.pi f₀` is
homotopic to `ContinuousMap.pi f₁`. |
protected piMap {X Y : ι → Type*} [∀ i, TopologicalSpace (X i)]
[∀ i, TopologicalSpace (Y i)] {f₀ f₁ : ∀ i, C(X i, Y i)} (F : ∀ i, Homotopic (f₀ i) (f₁ i)) :
Homotopic (.piMap f₀) (.piMap f₁) :=
.pi fun i ↦ .comp (F i) (.refl <| .eval i) | theorem | Topology | [
"Mathlib.Topology.Order.ProjIcc",
"Mathlib.Topology.ContinuousMap.Ordered",
"Mathlib.Topology.CompactOpen",
"Mathlib.Topology.UnitInterval"
] | Mathlib/Topology/Homotopy/Basic.lean | piMap | If each `f₀ i : C(X, Y i)` is homotopic to `f₁ i : C(X, Y i)`, then `ContinuousMap.pi f₀` is
homotopic to `ContinuousMap.pi f₁`. |
HomotopyWith (f₀ f₁ : C(X, Y)) (P : C(X, Y) → Prop) extends Homotopy f₀ f₁ where
/-- the intermediate maps of the homotopy satisfy the property -/
prop' : ∀ t, P ⟨fun x => toFun (t, x),
Continuous.comp continuous_toFun (continuous_const.prodMk continuous_id')⟩ | structure | Topology | [
"Mathlib.Topology.Order.ProjIcc",
"Mathlib.Topology.ContinuousMap.Ordered",
"Mathlib.Topology.CompactOpen",
"Mathlib.Topology.UnitInterval"
] | Mathlib/Topology/Homotopy/Basic.lean | HomotopyWith | The type of homotopies between `f₀ f₁ : C(X, Y)`, where the intermediate maps satisfy the predicate
`P : C(X, Y) → Prop` |
instFunLike : FunLike (HomotopyWith f₀ f₁ P) (I × X) Y where
coe F := ⇑F.toHomotopy
coe_injective'
| ⟨⟨⟨_, _⟩, _, _⟩, _⟩, ⟨⟨⟨_, _⟩, _, _⟩, _⟩, rfl => rfl | instance | Topology | [
"Mathlib.Topology.Order.ProjIcc",
"Mathlib.Topology.ContinuousMap.Ordered",
"Mathlib.Topology.CompactOpen",
"Mathlib.Topology.UnitInterval"
] | Mathlib/Topology/Homotopy/Basic.lean | instFunLike | null |
coeFn_injective : @Function.Injective (HomotopyWith f₀ f₁ P) (I × X → Y) (⇑) :=
DFunLike.coe_injective'
@[ext] | theorem | Topology | [
"Mathlib.Topology.Order.ProjIcc",
"Mathlib.Topology.ContinuousMap.Ordered",
"Mathlib.Topology.CompactOpen",
"Mathlib.Topology.UnitInterval"
] | Mathlib/Topology/Homotopy/Basic.lean | coeFn_injective | null |
ext {F G : HomotopyWith f₀ f₁ P} (h : ∀ x, F x = G x) : F = G := DFunLike.ext F G h | theorem | Topology | [
"Mathlib.Topology.Order.ProjIcc",
"Mathlib.Topology.ContinuousMap.Ordered",
"Mathlib.Topology.CompactOpen",
"Mathlib.Topology.UnitInterval"
] | Mathlib/Topology/Homotopy/Basic.lean | ext | null |
Simps.apply (F : HomotopyWith f₀ f₁ P) : I × X → Y := F
initialize_simps_projections HomotopyWith (toFun → apply, -toHomotopy_toContinuousMap)
@[continuity] | def | Topology | [
"Mathlib.Topology.Order.ProjIcc",
"Mathlib.Topology.ContinuousMap.Ordered",
"Mathlib.Topology.CompactOpen",
"Mathlib.Topology.UnitInterval"
] | Mathlib/Topology/Homotopy/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 continuous (F : HomotopyWith f₀ f₁ P) : Continuous F :=
F.continuous_toFun
@[simp] | theorem | Topology | [
"Mathlib.Topology.Order.ProjIcc",
"Mathlib.Topology.ContinuousMap.Ordered",
"Mathlib.Topology.CompactOpen",
"Mathlib.Topology.UnitInterval"
] | Mathlib/Topology/Homotopy/Basic.lean | continuous | null |
apply_zero (F : HomotopyWith f₀ f₁ P) (x : X) : F (0, x) = f₀ x :=
F.map_zero_left x
@[simp] | theorem | Topology | [
"Mathlib.Topology.Order.ProjIcc",
"Mathlib.Topology.ContinuousMap.Ordered",
"Mathlib.Topology.CompactOpen",
"Mathlib.Topology.UnitInterval"
] | Mathlib/Topology/Homotopy/Basic.lean | apply_zero | null |
apply_one (F : HomotopyWith f₀ f₁ P) (x : X) : F (1, x) = f₁ x :=
F.map_one_left x | theorem | Topology | [
"Mathlib.Topology.Order.ProjIcc",
"Mathlib.Topology.ContinuousMap.Ordered",
"Mathlib.Topology.CompactOpen",
"Mathlib.Topology.UnitInterval"
] | Mathlib/Topology/Homotopy/Basic.lean | apply_one | null |
coe_toContinuousMap (F : HomotopyWith f₀ f₁ P) : ⇑F.toContinuousMap = F :=
rfl
@[simp] | theorem | Topology | [
"Mathlib.Topology.Order.ProjIcc",
"Mathlib.Topology.ContinuousMap.Ordered",
"Mathlib.Topology.CompactOpen",
"Mathlib.Topology.UnitInterval"
] | Mathlib/Topology/Homotopy/Basic.lean | coe_toContinuousMap | null |
coe_toHomotopy (F : HomotopyWith f₀ f₁ P) : ⇑F.toHomotopy = F :=
rfl | theorem | Topology | [
"Mathlib.Topology.Order.ProjIcc",
"Mathlib.Topology.ContinuousMap.Ordered",
"Mathlib.Topology.CompactOpen",
"Mathlib.Topology.UnitInterval"
] | Mathlib/Topology/Homotopy/Basic.lean | coe_toHomotopy | null |
prop (F : HomotopyWith f₀ f₁ P) (t : I) : P (F.toHomotopy.curry t) := F.prop' t | theorem | Topology | [
"Mathlib.Topology.Order.ProjIcc",
"Mathlib.Topology.ContinuousMap.Ordered",
"Mathlib.Topology.CompactOpen",
"Mathlib.Topology.UnitInterval"
] | Mathlib/Topology/Homotopy/Basic.lean | prop | null |
extendProp (F : HomotopyWith f₀ f₁ P) (t : ℝ) : P (F.toHomotopy.extend t) := F.prop _ | theorem | Topology | [
"Mathlib.Topology.Order.ProjIcc",
"Mathlib.Topology.ContinuousMap.Ordered",
"Mathlib.Topology.CompactOpen",
"Mathlib.Topology.UnitInterval"
] | Mathlib/Topology/Homotopy/Basic.lean | extendProp | null |
@[simps!]
refl (f : C(X, Y)) (hf : P f) : HomotopyWith f f P where
toHomotopy := Homotopy.refl f
prop' := fun _ => hf | def | Topology | [
"Mathlib.Topology.Order.ProjIcc",
"Mathlib.Topology.ContinuousMap.Ordered",
"Mathlib.Topology.CompactOpen",
"Mathlib.Topology.UnitInterval"
] | Mathlib/Topology/Homotopy/Basic.lean | refl | Given a continuous function `f`, and a proof `h : P f`, we can define a `HomotopyWith f f P` by
`F (t, x) = f x` |
@[simps!]
symm {f₀ f₁ : C(X, Y)} (F : HomotopyWith f₀ f₁ P) : HomotopyWith f₁ f₀ P where
toHomotopy := F.toHomotopy.symm
prop' := fun t => F.prop (σ t)
@[simp] | def | Topology | [
"Mathlib.Topology.Order.ProjIcc",
"Mathlib.Topology.ContinuousMap.Ordered",
"Mathlib.Topology.CompactOpen",
"Mathlib.Topology.UnitInterval"
] | Mathlib/Topology/Homotopy/Basic.lean | symm | Given a `HomotopyWith f₀ f₁ P`, we can define a `HomotopyWith f₁ f₀ P` by reversing the homotopy. |
symm_symm {f₀ f₁ : C(X, Y)} (F : HomotopyWith f₀ f₁ P) : F.symm.symm = F :=
ext <| Homotopy.congr_fun <| Homotopy.symm_symm _ | theorem | Topology | [
"Mathlib.Topology.Order.ProjIcc",
"Mathlib.Topology.ContinuousMap.Ordered",
"Mathlib.Topology.CompactOpen",
"Mathlib.Topology.UnitInterval"
] | Mathlib/Topology/Homotopy/Basic.lean | symm_symm | null |
symm_bijective {f₀ f₁ : C(X, Y)} :
Function.Bijective (HomotopyWith.symm : HomotopyWith f₀ f₁ P → HomotopyWith f₁ f₀ P) :=
Function.bijective_iff_has_inverse.mpr ⟨_, symm_symm, symm_symm⟩ | theorem | Topology | [
"Mathlib.Topology.Order.ProjIcc",
"Mathlib.Topology.ContinuousMap.Ordered",
"Mathlib.Topology.CompactOpen",
"Mathlib.Topology.UnitInterval"
] | Mathlib/Topology/Homotopy/Basic.lean | symm_bijective | null |
trans {f₀ f₁ f₂ : C(X, Y)} (F : HomotopyWith f₀ f₁ P) (G : HomotopyWith f₁ f₂ P) :
HomotopyWith f₀ f₂ P :=
{ F.toHomotopy.trans G.toHomotopy with
prop' := fun t => by
simp only [Homotopy.trans]
change P ⟨fun _ => ite ((t : ℝ) ≤ _) _ _, _⟩
split_ifs
· exact F.extendProp _
· exact G.extendProp _ } | def | Topology | [
"Mathlib.Topology.Order.ProjIcc",
"Mathlib.Topology.ContinuousMap.Ordered",
"Mathlib.Topology.CompactOpen",
"Mathlib.Topology.UnitInterval"
] | Mathlib/Topology/Homotopy/Basic.lean | trans | Given `HomotopyWith f₀ f₁ P` and `HomotopyWith f₁ f₂ P`, we can define a `HomotopyWith f₀ f₂ P`
by putting the first homotopy on `[0, 1/2]` and the second on `[1/2, 1]`. |
trans_apply {f₀ f₁ f₂ : C(X, Y)} (F : HomotopyWith f₀ f₁ P) (G : HomotopyWith f₁ f₂ P)
(x : I × X) :
(F.trans G) x =
if h : (x.1 : ℝ) ≤ 1 / 2 then
F (⟨2 * x.1, (unitInterval.mul_pos_mem_iff zero_lt_two).2 ⟨x.1.2.1, h⟩⟩, x.2)
else
G (⟨2 * x.1 - 1, unitInterval.two_mul_sub_one_mem_iff.2 ⟨(not_le.1 h).le, x.1.2.2⟩⟩, x.2) :=
Homotopy.trans_apply _ _ _ | theorem | Topology | [
"Mathlib.Topology.Order.ProjIcc",
"Mathlib.Topology.ContinuousMap.Ordered",
"Mathlib.Topology.CompactOpen",
"Mathlib.Topology.UnitInterval"
] | Mathlib/Topology/Homotopy/Basic.lean | trans_apply | null |
symm_trans {f₀ f₁ f₂ : C(X, Y)} (F : HomotopyWith f₀ f₁ P) (G : HomotopyWith f₁ f₂ P) :
(F.trans G).symm = G.symm.trans F.symm :=
ext <| Homotopy.congr_fun <| Homotopy.symm_trans _ _ | theorem | Topology | [
"Mathlib.Topology.Order.ProjIcc",
"Mathlib.Topology.ContinuousMap.Ordered",
"Mathlib.Topology.CompactOpen",
"Mathlib.Topology.UnitInterval"
] | Mathlib/Topology/Homotopy/Basic.lean | symm_trans | null |
@[simps!]
cast {f₀ f₁ g₀ g₁ : C(X, Y)} (F : HomotopyWith f₀ f₁ P) (h₀ : f₀ = g₀) (h₁ : f₁ = g₁) :
HomotopyWith g₀ g₁ P where
toHomotopy := F.toHomotopy.cast h₀ h₁
prop' := F.prop | def | Topology | [
"Mathlib.Topology.Order.ProjIcc",
"Mathlib.Topology.ContinuousMap.Ordered",
"Mathlib.Topology.CompactOpen",
"Mathlib.Topology.UnitInterval"
] | Mathlib/Topology/Homotopy/Basic.lean | cast | Casting a `HomotopyWith f₀ f₁ P` to a `HomotopyWith g₀ g₁ P` where `f₀ = g₀` and `f₁ = g₁`. |
HomotopicWith (f₀ f₁ : C(X, Y)) (P : C(X, Y) → Prop) : Prop :=
Nonempty (HomotopyWith f₀ f₁ P) | def | Topology | [
"Mathlib.Topology.Order.ProjIcc",
"Mathlib.Topology.ContinuousMap.Ordered",
"Mathlib.Topology.CompactOpen",
"Mathlib.Topology.UnitInterval"
] | Mathlib/Topology/Homotopy/Basic.lean | HomotopicWith | Given continuous maps `f₀` and `f₁`, we say `f₀` and `f₁` are homotopic with respect to the
predicate `P` if there exists a `HomotopyWith f₀ f₁ P`. |
refl (f : C(X, Y)) (hf : P f) : HomotopicWith f f P :=
⟨HomotopyWith.refl f hf⟩
@[symm] | theorem | Topology | [
"Mathlib.Topology.Order.ProjIcc",
"Mathlib.Topology.ContinuousMap.Ordered",
"Mathlib.Topology.CompactOpen",
"Mathlib.Topology.UnitInterval"
] | Mathlib/Topology/Homotopy/Basic.lean | refl | null |
symm ⦃f g : C(X, Y)⦄ (h : HomotopicWith f g P) : HomotopicWith g f P :=
⟨h.some.symm⟩
@[trans] | theorem | Topology | [
"Mathlib.Topology.Order.ProjIcc",
"Mathlib.Topology.ContinuousMap.Ordered",
"Mathlib.Topology.CompactOpen",
"Mathlib.Topology.UnitInterval"
] | Mathlib/Topology/Homotopy/Basic.lean | symm | null |
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