fact stringlengths 6 3.84k | type stringclasses 11 values | library stringclasses 32 values | imports listlengths 1 14 | filename stringlengths 20 95 | symbolic_name stringlengths 1 90 | docstring stringlengths 7 20k ⌀ |
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preimage_symm (h : X ≃ₜ Y) : preimage h.symm = image h :=
(funext h.toEquiv.image_eq_preimage).symm
@[simp] | theorem | Topology | [
"Mathlib.Topology.ContinuousMap.Defs",
"Mathlib.Topology.Maps.Basic"
] | Mathlib/Topology/Homeomorph/Defs.lean | preimage_symm | null |
image_preimage (h : X ≃ₜ Y) (s : Set Y) : h '' (h ⁻¹' s) = s :=
h.toEquiv.image_preimage s
@[simp] | theorem | Topology | [
"Mathlib.Topology.ContinuousMap.Defs",
"Mathlib.Topology.Maps.Basic"
] | Mathlib/Topology/Homeomorph/Defs.lean | image_preimage | null |
preimage_image (h : X ≃ₜ Y) (s : Set X) : h ⁻¹' (h '' s) = s :=
h.toEquiv.preimage_image s | theorem | Topology | [
"Mathlib.Topology.ContinuousMap.Defs",
"Mathlib.Topology.Maps.Basic"
] | Mathlib/Topology/Homeomorph/Defs.lean | preimage_image | null |
image_eq_preimage (h : X ≃ₜ Y) (s : Set X) : h '' s = h.symm ⁻¹' s :=
h.toEquiv.image_eq_preimage s | theorem | Topology | [
"Mathlib.Topology.ContinuousMap.Defs",
"Mathlib.Topology.Maps.Basic"
] | Mathlib/Topology/Homeomorph/Defs.lean | image_eq_preimage | null |
image_compl (h : X ≃ₜ Y) (s : Set X) : h '' (sᶜ) = (h '' s)ᶜ :=
h.toEquiv.image_compl s | lemma | Topology | [
"Mathlib.Topology.ContinuousMap.Defs",
"Mathlib.Topology.Maps.Basic"
] | Mathlib/Topology/Homeomorph/Defs.lean | image_compl | null |
isInducing (h : X ≃ₜ Y) : IsInducing h :=
.of_comp h.continuous h.symm.continuous <| by simp only [symm_comp_self, IsInducing.id] | lemma | Topology | [
"Mathlib.Topology.ContinuousMap.Defs",
"Mathlib.Topology.Maps.Basic"
] | Mathlib/Topology/Homeomorph/Defs.lean | isInducing | null |
induced_eq (h : X ≃ₜ Y) : TopologicalSpace.induced h ‹_› = ‹_› := h.isInducing.1.symm | theorem | Topology | [
"Mathlib.Topology.ContinuousMap.Defs",
"Mathlib.Topology.Maps.Basic"
] | Mathlib/Topology/Homeomorph/Defs.lean | induced_eq | null |
isQuotientMap (h : X ≃ₜ Y) : IsQuotientMap h :=
IsQuotientMap.of_comp h.symm.continuous h.continuous <| by
simp only [self_comp_symm, IsQuotientMap.id] | theorem | Topology | [
"Mathlib.Topology.ContinuousMap.Defs",
"Mathlib.Topology.Maps.Basic"
] | Mathlib/Topology/Homeomorph/Defs.lean | isQuotientMap | null |
coinduced_eq (h : X ≃ₜ Y) : TopologicalSpace.coinduced h ‹_› = ‹_› :=
h.isQuotientMap.2.symm | theorem | Topology | [
"Mathlib.Topology.ContinuousMap.Defs",
"Mathlib.Topology.Maps.Basic"
] | Mathlib/Topology/Homeomorph/Defs.lean | coinduced_eq | null |
isEmbedding (h : X ≃ₜ Y) : IsEmbedding h := ⟨h.isInducing, h.injective⟩ | theorem | Topology | [
"Mathlib.Topology.ContinuousMap.Defs",
"Mathlib.Topology.Maps.Basic"
] | Mathlib/Topology/Homeomorph/Defs.lean | isEmbedding | null |
protected discreteTopology [DiscreteTopology X] (h : X ≃ₜ Y) : DiscreteTopology Y :=
h.symm.isEmbedding.discreteTopology | theorem | Topology | [
"Mathlib.Topology.ContinuousMap.Defs",
"Mathlib.Topology.Maps.Basic"
] | Mathlib/Topology/Homeomorph/Defs.lean | discreteTopology | null |
discreteTopology_iff (h : X ≃ₜ Y) : DiscreteTopology X ↔ DiscreteTopology Y :=
⟨fun _ ↦ h.discreteTopology, fun _ ↦ h.symm.discreteTopology⟩
@[simp] | theorem | Topology | [
"Mathlib.Topology.ContinuousMap.Defs",
"Mathlib.Topology.Maps.Basic"
] | Mathlib/Topology/Homeomorph/Defs.lean | discreteTopology_iff | null |
isOpen_preimage (h : X ≃ₜ Y) {s : Set Y} : IsOpen (h ⁻¹' s) ↔ IsOpen s :=
h.isQuotientMap.isOpen_preimage
@[simp] | theorem | Topology | [
"Mathlib.Topology.ContinuousMap.Defs",
"Mathlib.Topology.Maps.Basic"
] | Mathlib/Topology/Homeomorph/Defs.lean | isOpen_preimage | null |
isOpen_image (h : X ≃ₜ Y) {s : Set X} : IsOpen (h '' s) ↔ IsOpen s := by
rw [← preimage_symm, isOpen_preimage] | theorem | Topology | [
"Mathlib.Topology.ContinuousMap.Defs",
"Mathlib.Topology.Maps.Basic"
] | Mathlib/Topology/Homeomorph/Defs.lean | isOpen_image | null |
protected isOpenMap (h : X ≃ₜ Y) : IsOpenMap h := fun _ => h.isOpen_image.2 | theorem | Topology | [
"Mathlib.Topology.ContinuousMap.Defs",
"Mathlib.Topology.Maps.Basic"
] | Mathlib/Topology/Homeomorph/Defs.lean | isOpenMap | null |
protected isOpenQuotientMap (h : X ≃ₜ Y) : IsOpenQuotientMap h :=
⟨h.surjective, h.continuous, h.isOpenMap⟩
@[simp] | theorem | Topology | [
"Mathlib.Topology.ContinuousMap.Defs",
"Mathlib.Topology.Maps.Basic"
] | Mathlib/Topology/Homeomorph/Defs.lean | isOpenQuotientMap | null |
isClosed_preimage (h : X ≃ₜ Y) {s : Set Y} : IsClosed (h ⁻¹' s) ↔ IsClosed s := by
simp only [← isOpen_compl_iff, ← preimage_compl, isOpen_preimage]
@[simp] | theorem | Topology | [
"Mathlib.Topology.ContinuousMap.Defs",
"Mathlib.Topology.Maps.Basic"
] | Mathlib/Topology/Homeomorph/Defs.lean | isClosed_preimage | null |
isClosed_image (h : X ≃ₜ Y) {s : Set X} : IsClosed (h '' s) ↔ IsClosed s := by
rw [← preimage_symm, isClosed_preimage] | theorem | Topology | [
"Mathlib.Topology.ContinuousMap.Defs",
"Mathlib.Topology.Maps.Basic"
] | Mathlib/Topology/Homeomorph/Defs.lean | isClosed_image | null |
protected isClosedMap (h : X ≃ₜ Y) : IsClosedMap h := fun _ => h.isClosed_image.2 | theorem | Topology | [
"Mathlib.Topology.ContinuousMap.Defs",
"Mathlib.Topology.Maps.Basic"
] | Mathlib/Topology/Homeomorph/Defs.lean | isClosedMap | null |
isOpenEmbedding (h : X ≃ₜ Y) : IsOpenEmbedding h :=
.of_isEmbedding_isOpenMap h.isEmbedding h.isOpenMap | theorem | Topology | [
"Mathlib.Topology.ContinuousMap.Defs",
"Mathlib.Topology.Maps.Basic"
] | Mathlib/Topology/Homeomorph/Defs.lean | isOpenEmbedding | null |
isClosedEmbedding (h : X ≃ₜ Y) : IsClosedEmbedding h :=
.of_isEmbedding_isClosedMap h.isEmbedding h.isClosedMap | theorem | Topology | [
"Mathlib.Topology.ContinuousMap.Defs",
"Mathlib.Topology.Maps.Basic"
] | Mathlib/Topology/Homeomorph/Defs.lean | isClosedEmbedding | null |
preimage_closure (h : X ≃ₜ Y) (s : Set Y) : h ⁻¹' closure s = closure (h ⁻¹' s) :=
h.isOpenMap.preimage_closure_eq_closure_preimage h.continuous _ | theorem | Topology | [
"Mathlib.Topology.ContinuousMap.Defs",
"Mathlib.Topology.Maps.Basic"
] | Mathlib/Topology/Homeomorph/Defs.lean | preimage_closure | null |
image_closure (h : X ≃ₜ Y) (s : Set X) : h '' closure s = closure (h '' s) := by
rw [← preimage_symm, preimage_closure] | theorem | Topology | [
"Mathlib.Topology.ContinuousMap.Defs",
"Mathlib.Topology.Maps.Basic"
] | Mathlib/Topology/Homeomorph/Defs.lean | image_closure | null |
preimage_interior (h : X ≃ₜ Y) (s : Set Y) : h ⁻¹' interior s = interior (h ⁻¹' s) :=
h.isOpenMap.preimage_interior_eq_interior_preimage h.continuous _ | theorem | Topology | [
"Mathlib.Topology.ContinuousMap.Defs",
"Mathlib.Topology.Maps.Basic"
] | Mathlib/Topology/Homeomorph/Defs.lean | preimage_interior | null |
image_interior (h : X ≃ₜ Y) (s : Set X) : h '' interior s = interior (h '' s) := by
rw [← preimage_symm, preimage_interior] | theorem | Topology | [
"Mathlib.Topology.ContinuousMap.Defs",
"Mathlib.Topology.Maps.Basic"
] | Mathlib/Topology/Homeomorph/Defs.lean | image_interior | null |
preimage_frontier (h : X ≃ₜ Y) (s : Set Y) : h ⁻¹' frontier s = frontier (h ⁻¹' s) :=
h.isOpenMap.preimage_frontier_eq_frontier_preimage h.continuous _ | theorem | Topology | [
"Mathlib.Topology.ContinuousMap.Defs",
"Mathlib.Topology.Maps.Basic"
] | Mathlib/Topology/Homeomorph/Defs.lean | preimage_frontier | null |
image_frontier (h : X ≃ₜ Y) (s : Set X) : h '' frontier s = frontier (h '' s) := by
rw [← preimage_symm, preimage_frontier]
@[simp] | theorem | Topology | [
"Mathlib.Topology.ContinuousMap.Defs",
"Mathlib.Topology.Maps.Basic"
] | Mathlib/Topology/Homeomorph/Defs.lean | image_frontier | null |
comp_continuous_iff (h : X ≃ₜ Y) {f : Z → X} : Continuous (h ∘ f) ↔ Continuous f :=
h.isInducing.continuous_iff.symm
@[simp] | theorem | Topology | [
"Mathlib.Topology.ContinuousMap.Defs",
"Mathlib.Topology.Maps.Basic"
] | Mathlib/Topology/Homeomorph/Defs.lean | comp_continuous_iff | null |
comp_continuous_iff' (h : X ≃ₜ Y) {f : Y → Z} : Continuous (f ∘ h) ↔ Continuous f :=
h.isQuotientMap.continuous_iff.symm | theorem | Topology | [
"Mathlib.Topology.ContinuousMap.Defs",
"Mathlib.Topology.Maps.Basic"
] | Mathlib/Topology/Homeomorph/Defs.lean | comp_continuous_iff' | null |
comp_continuousAt_iff (h : X ≃ₜ Y) (f : Z → X) (z : Z) :
ContinuousAt (h ∘ f) z ↔ ContinuousAt f z :=
h.isInducing.continuousAt_iff.symm | theorem | Topology | [
"Mathlib.Topology.ContinuousMap.Defs",
"Mathlib.Topology.Maps.Basic"
] | Mathlib/Topology/Homeomorph/Defs.lean | comp_continuousAt_iff | null |
comp_continuousAt_iff' (h : X ≃ₜ Y) (f : Y → Z) (x : X) :
ContinuousAt (f ∘ h) x ↔ ContinuousAt f (h x) :=
h.isInducing.continuousAt_iff' (by simp)
@[simp] | theorem | Topology | [
"Mathlib.Topology.ContinuousMap.Defs",
"Mathlib.Topology.Maps.Basic"
] | Mathlib/Topology/Homeomorph/Defs.lean | comp_continuousAt_iff' | null |
comp_isOpenMap_iff (h : X ≃ₜ Y) {f : Z → X} : IsOpenMap (h ∘ f) ↔ IsOpenMap f := by
refine ⟨?_, fun hf => h.isOpenMap.comp hf⟩
intro hf
rw [← Function.id_comp f, ← h.symm_comp_self, Function.comp_assoc]
exact h.symm.isOpenMap.comp hf
@[simp] | theorem | Topology | [
"Mathlib.Topology.ContinuousMap.Defs",
"Mathlib.Topology.Maps.Basic"
] | Mathlib/Topology/Homeomorph/Defs.lean | comp_isOpenMap_iff | null |
comp_isOpenMap_iff' (h : X ≃ₜ Y) {f : Y → Z} : IsOpenMap (f ∘ h) ↔ IsOpenMap f := by
refine ⟨?_, fun hf => hf.comp h.isOpenMap⟩
intro hf
rw [← Function.comp_id f, ← h.self_comp_symm, ← Function.comp_assoc]
exact hf.comp h.symm.isOpenMap
variable (X Y) in | theorem | Topology | [
"Mathlib.Topology.ContinuousMap.Defs",
"Mathlib.Topology.Maps.Basic"
] | Mathlib/Topology/Homeomorph/Defs.lean | comp_isOpenMap_iff' | null |
@[simps!]
homeomorphOfUnique [Unique X] [Unique Y] : X ≃ₜ Y :=
{ Equiv.ofUnique X Y with
continuous_toFun := continuous_const
continuous_invFun := continuous_const }
@[simp] | def | Topology | [
"Mathlib.Topology.ContinuousMap.Defs",
"Mathlib.Topology.Maps.Basic"
] | Mathlib/Topology/Homeomorph/Defs.lean | homeomorphOfUnique | If both `X` and `Y` have a unique element, then `X ≃ₜ Y`. |
map_nhds_eq (h : X ≃ₜ Y) (x : X) : map h (𝓝 x) = 𝓝 (h x) :=
h.isEmbedding.map_nhds_of_mem _ (by simp) | theorem | Topology | [
"Mathlib.Topology.ContinuousMap.Defs",
"Mathlib.Topology.Maps.Basic"
] | Mathlib/Topology/Homeomorph/Defs.lean | map_nhds_eq | null |
symm_map_nhds_eq (h : X ≃ₜ Y) (x : X) : map h.symm (𝓝 (h x)) = 𝓝 x := by
rw [h.symm.map_nhds_eq, h.symm_apply_apply] | theorem | Topology | [
"Mathlib.Topology.ContinuousMap.Defs",
"Mathlib.Topology.Maps.Basic"
] | Mathlib/Topology/Homeomorph/Defs.lean | symm_map_nhds_eq | null |
nhds_eq_comap (h : X ≃ₜ Y) (x : X) : 𝓝 x = comap h (𝓝 (h x)) :=
h.isInducing.nhds_eq_comap x
@[simp] | theorem | Topology | [
"Mathlib.Topology.ContinuousMap.Defs",
"Mathlib.Topology.Maps.Basic"
] | Mathlib/Topology/Homeomorph/Defs.lean | nhds_eq_comap | null |
comap_nhds_eq (h : X ≃ₜ Y) (y : Y) : comap h (𝓝 y) = 𝓝 (h.symm y) := by
rw [h.nhds_eq_comap, h.apply_symm_apply] | theorem | Topology | [
"Mathlib.Topology.ContinuousMap.Defs",
"Mathlib.Topology.Maps.Basic"
] | Mathlib/Topology/Homeomorph/Defs.lean | comap_nhds_eq | null |
@[simps toEquiv]
toHomeomorph (e : X ≃ Y) (he : ∀ s, IsOpen (e ⁻¹' s) ↔ IsOpen s) : X ≃ₜ Y where
toEquiv := e
continuous_toFun := continuous_def.2 fun _ ↦ (he _).2
continuous_invFun := continuous_def.2 fun s ↦ by convert (he _).1; simp
@[simp] lemma coe_toHomeomorph (e : X ≃ Y) (he) : ⇑(e.toHomeomorph he) = e := rfl | def | Topology | [
"Mathlib.Topology.ContinuousMap.Defs",
"Mathlib.Topology.Maps.Basic"
] | Mathlib/Topology/Homeomorph/Defs.lean | toHomeomorph | An equivalence between topological spaces respecting openness is a homeomorphism. |
toHomeomorph_apply (e : X ≃ Y) (he) (x : X) : e.toHomeomorph he x = e x := rfl
@[simp] lemma toHomeomorph_refl :
(Equiv.refl X).toHomeomorph (fun _s ↦ Iff.rfl) = Homeomorph.refl _ := rfl
@[simp] lemma toHomeomorph_symm (e : X ≃ Y) (he) :
(e.toHomeomorph he).symm = e.symm.toHomeomorph fun s ↦ by convert (he _).symm; simp := rfl | lemma | Topology | [
"Mathlib.Topology.ContinuousMap.Defs",
"Mathlib.Topology.Maps.Basic"
] | Mathlib/Topology/Homeomorph/Defs.lean | toHomeomorph_apply | null |
toHomeomorph_trans (e : X ≃ Y) (f : Y ≃ Z) (he hf) :
(e.trans f).toHomeomorph (fun _s ↦ (he _).trans (hf _)) =
(e.toHomeomorph he).trans (f.toHomeomorph hf) := rfl | lemma | Topology | [
"Mathlib.Topology.ContinuousMap.Defs",
"Mathlib.Topology.Maps.Basic"
] | Mathlib/Topology/Homeomorph/Defs.lean | toHomeomorph_trans | null |
@[simps toEquiv]
toHomeomorphOfIsInducing (f : X ≃ Y) (hf : IsInducing f) : X ≃ₜ Y :=
{ f with
continuous_toFun := hf.continuous
continuous_invFun := hf.continuous_iff.2 <| by simpa using continuous_id }
@[simp] lemma toHomeomorphOfIsInducing_apply (f : X ≃ Y) (hf : IsInducing f) :
⇑(f.toHomeomorphOfIsInducing hf) = f := rfl
@[simp] lemma toHomeomorphOfIsInducing_symm_apply (f : X ≃ Y) (hf : IsInducing f) :
⇑(f.toHomeomorphOfIsInducing hf).symm = f.symm := rfl | def | Topology | [
"Mathlib.Topology.ContinuousMap.Defs",
"Mathlib.Topology.Maps.Basic"
] | Mathlib/Topology/Homeomorph/Defs.lean | toHomeomorphOfIsInducing | An inducing equiv between topological spaces is a homeomorphism. |
@[simps! toEquiv]
toHomeomorphOfContinuousOpen (e : X ≃ Y) (h₁ : Continuous e) (h₂ : IsOpenMap e) : X ≃ₜ Y :=
e.toHomeomorphOfIsInducing <|
IsOpenEmbedding.of_continuous_injective_isOpenMap h₁ e.injective h₂ |>.toIsInducing
@[deprecated (since := "2025-04-16")]
alias _root_.Homeomorph.homeomorphOfContinuousOpen := toHomeomorphOfContinuousOpen
@[deprecated (since := "2025-04-16")]
alias _root_.Homeomorph.homeomorphOfContinuousOpen_toEquiv := toHomeomorphOfContinuousOpen_toEquiv
@[simp] | def | Topology | [
"Mathlib.Topology.ContinuousMap.Defs",
"Mathlib.Topology.Maps.Basic"
] | Mathlib/Topology/Homeomorph/Defs.lean | toHomeomorphOfContinuousOpen | If a bijective map `e : X ≃ Y` is continuous and open, then it is a homeomorphism. |
toHomeomorphOfContinuousOpen_apply (e : X ≃ Y) (h₁ : Continuous e) (h₂ : IsOpenMap e) :
⇑(e.toHomeomorphOfContinuousOpen h₁ h₂) = e := rfl
@[deprecated (since := "2025-04-16")]
alias _root_.Homeomorph.homeomorphOfContinuousOpen_apply := toHomeomorphOfContinuousOpen_apply
@[simp] | theorem | Topology | [
"Mathlib.Topology.ContinuousMap.Defs",
"Mathlib.Topology.Maps.Basic"
] | Mathlib/Topology/Homeomorph/Defs.lean | toHomeomorphOfContinuousOpen_apply | null |
toHomeomorphOfContinuousOpen_symm_apply (e : X ≃ Y) (h₁ : Continuous e) (h₂ : IsOpenMap e) :
⇑(e.toHomeomorphOfContinuousOpen h₁ h₂).symm = e.symm := rfl
@[deprecated (since := "2025-04-16")]
alias _root_.Homeomorph.homeomorphOfContinuousOpen_symm_apply :=
toHomeomorphOfContinuousOpen_symm_apply | theorem | Topology | [
"Mathlib.Topology.ContinuousMap.Defs",
"Mathlib.Topology.Maps.Basic"
] | Mathlib/Topology/Homeomorph/Defs.lean | toHomeomorphOfContinuousOpen_symm_apply | null |
@[simps! toEquiv]
toHomeomorphOfContinuousClosed (e : X ≃ Y) (h₁ : Continuous e) (h₂ : IsClosedMap e) : X ≃ₜ Y :=
e.toHomeomorphOfIsInducing <|
IsClosedEmbedding.of_continuous_injective_isClosedMap h₁ e.injective h₂ |>.toIsInducing
@[deprecated (since := "2025-04-16")]
alias _root_.Homeomorph.homeomorphOfContinuousClosed := toHomeomorphOfContinuousClosed
@[simp] | def | Topology | [
"Mathlib.Topology.ContinuousMap.Defs",
"Mathlib.Topology.Maps.Basic"
] | Mathlib/Topology/Homeomorph/Defs.lean | toHomeomorphOfContinuousClosed | If a bijective map `e : X ≃ Y` is continuous and open, then it is a homeomorphism. |
toHomeomorphOfContinuousClosed_apply (e : X ≃ Y) (h₁ : Continuous e) (h₂ : IsClosedMap e) :
⇑(e.toHomeomorphOfContinuousClosed h₁ h₂) = e := rfl
@[simp] | theorem | Topology | [
"Mathlib.Topology.ContinuousMap.Defs",
"Mathlib.Topology.Maps.Basic"
] | Mathlib/Topology/Homeomorph/Defs.lean | toHomeomorphOfContinuousClosed_apply | null |
toHomeomorphOfContinuousClosed_symm_apply
(e : X ≃ Y) (h₁ : Continuous e) (h₂ : IsClosedMap e) :
⇑(e.toHomeomorphOfContinuousClosed h₁ h₂).symm = e.symm := rfl | theorem | Topology | [
"Mathlib.Topology.ContinuousMap.Defs",
"Mathlib.Topology.Maps.Basic"
] | Mathlib/Topology/Homeomorph/Defs.lean | toHomeomorphOfContinuousClosed_symm_apply | null |
HomeomorphClass (F : Type*) (A B : outParam Type*)
[TopologicalSpace A] [TopologicalSpace B] [h : EquivLike F A B] : Prop where
map_continuous : ∀ (f : F), Continuous f
inv_continuous : ∀ (f : F), Continuous (h.inv f) | class | Topology | [
"Mathlib.Topology.ContinuousMap.Defs",
"Mathlib.Topology.Maps.Basic"
] | Mathlib/Topology/Homeomorph/Defs.lean | HomeomorphClass | `HomeomorphClass F A B` states that `F` is a type of homeomorphisms. |
@[coe]
toHomeomorph [h : HomeomorphClass F α β] (f : F) : α ≃ₜ β :=
{ (f : α ≃ β) with
continuous_toFun := h.map_continuous f
continuous_invFun := h.inv_continuous f }
@[simp] | def | Topology | [
"Mathlib.Topology.ContinuousMap.Defs",
"Mathlib.Topology.Maps.Basic"
] | Mathlib/Topology/Homeomorph/Defs.lean | toHomeomorph | Turn an element of a type `F` satisfying `HomeomorphClass F α β` into an actual
`Homeomorph`. This is declared as the default coercion from `F` to `α ≃ₜ β`. |
coe_coe [h : HomeomorphClass F α β] (f : F) : ⇑(h.toHomeomorph f) = ⇑f := rfl | theorem | Topology | [
"Mathlib.Topology.ContinuousMap.Defs",
"Mathlib.Topology.Maps.Basic"
] | Mathlib/Topology/Homeomorph/Defs.lean | coe_coe | null |
toHomeomorph_injective [HomeomorphClass F α β] : Function.Injective ((↑) : F → α ≃ₜ β) :=
fun _ _ e ↦ DFunLike.ext _ _ fun a ↦ congr_arg (fun e : α ≃ₜ β ↦ e.toFun a) e | theorem | Topology | [
"Mathlib.Topology.ContinuousMap.Defs",
"Mathlib.Topology.Maps.Basic"
] | Mathlib/Topology/Homeomorph/Defs.lean | toHomeomorph_injective | null |
IsHomeomorph (f : X → Y) : Prop where
continuous : Continuous f
isOpenMap : IsOpenMap f
bijective : Function.Bijective f | structure | Topology | [
"Mathlib.Topology.ContinuousMap.Defs",
"Mathlib.Topology.Maps.Basic"
] | Mathlib/Topology/Homeomorph/Defs.lean | IsHomeomorph | Predicate saying that `f` is a homeomorphism.
This should be used only when `f` is a concrete function whose continuous inverse is not easy to
write down. Otherwise, `Homeomorph` should be preferred as it bundles the continuous inverse.
Having both `Homeomorph` and `IsHomeomorph` is justified by the fact that so many function
properties are unbundled in the topology part of the library, and by the fact that a homeomorphism
is not merely a continuous bijection, that is `IsHomeomorph f` is not equivalent to
`Continuous f ∧ Bijective f` but to `Continuous f ∧ Bijective f ∧ IsOpenMap f`. |
protected Homeomorph.isHomeomorph (h : X ≃ₜ Y) : IsHomeomorph h :=
⟨h.continuous, h.isOpenMap, h.bijective⟩ | theorem | Topology | [
"Mathlib.Topology.ContinuousMap.Defs",
"Mathlib.Topology.Maps.Basic"
] | Mathlib/Topology/Homeomorph/Defs.lean | Homeomorph.isHomeomorph | null |
protected injective (hf : IsHomeomorph f) : Function.Injective f := hf.bijective.injective | lemma | Topology | [
"Mathlib.Topology.ContinuousMap.Defs",
"Mathlib.Topology.Maps.Basic"
] | Mathlib/Topology/Homeomorph/Defs.lean | injective | null |
protected surjective (hf : IsHomeomorph f) : Function.Surjective f := hf.bijective.surjective | lemma | Topology | [
"Mathlib.Topology.ContinuousMap.Defs",
"Mathlib.Topology.Maps.Basic"
] | Mathlib/Topology/Homeomorph/Defs.lean | surjective | null |
protected id : IsHomeomorph (@id X) := ⟨continuous_id, .id, Function.bijective_id⟩ | lemma | Topology | [
"Mathlib.Topology.ContinuousMap.Defs",
"Mathlib.Topology.Maps.Basic"
] | Mathlib/Topology/Homeomorph/Defs.lean | id | null |
comp {g : Y → Z} (hg : IsHomeomorph g) (hf : IsHomeomorph f) : IsHomeomorph (g ∘ f) :=
⟨hg.1.comp hf.1, hg.2.comp hf.2, hg.3.comp hf.3⟩ | lemma | Topology | [
"Mathlib.Topology.ContinuousMap.Defs",
"Mathlib.Topology.Maps.Basic"
] | Mathlib/Topology/Homeomorph/Defs.lean | comp | null |
protected secondCountableTopology [SecondCountableTopology Y]
(h : X ≃ₜ Y) : SecondCountableTopology X :=
h.isInducing.secondCountableTopology | theorem | Topology | [
"Mathlib.Logic.Equiv.Fin.Basic",
"Mathlib.Topology.Connected.LocallyConnected",
"Mathlib.Topology.DenseEmbedding",
"Mathlib.Topology.Connected.TotallyDisconnected"
] | Mathlib/Topology/Homeomorph/Lemmas.lean | secondCountableTopology | null |
@[simp]
isCompact_image {s : Set X} (h : X ≃ₜ Y) : IsCompact (h '' s) ↔ IsCompact s :=
h.isEmbedding.isCompact_iff.symm | theorem | Topology | [
"Mathlib.Logic.Equiv.Fin.Basic",
"Mathlib.Topology.Connected.LocallyConnected",
"Mathlib.Topology.DenseEmbedding",
"Mathlib.Topology.Connected.TotallyDisconnected"
] | Mathlib/Topology/Homeomorph/Lemmas.lean | isCompact_image | If `h : X → Y` is a homeomorphism, `h(s)` is compact iff `s` is. |
@[simp]
isCompact_preimage {s : Set Y} (h : X ≃ₜ Y) : IsCompact (h ⁻¹' s) ↔ IsCompact s := by
rw [← image_symm]; exact h.symm.isCompact_image | theorem | Topology | [
"Mathlib.Logic.Equiv.Fin.Basic",
"Mathlib.Topology.Connected.LocallyConnected",
"Mathlib.Topology.DenseEmbedding",
"Mathlib.Topology.Connected.TotallyDisconnected"
] | Mathlib/Topology/Homeomorph/Lemmas.lean | isCompact_preimage | If `h : X → Y` is a homeomorphism, `h⁻¹(s)` is compact iff `s` is. |
@[simp]
isSigmaCompact_image {s : Set X} (h : X ≃ₜ Y) :
IsSigmaCompact (h '' s) ↔ IsSigmaCompact s :=
h.isEmbedding.isSigmaCompact_iff.symm | theorem | Topology | [
"Mathlib.Logic.Equiv.Fin.Basic",
"Mathlib.Topology.Connected.LocallyConnected",
"Mathlib.Topology.DenseEmbedding",
"Mathlib.Topology.Connected.TotallyDisconnected"
] | Mathlib/Topology/Homeomorph/Lemmas.lean | isSigmaCompact_image | If `h : X → Y` is a homeomorphism, `s` is σ-compact iff `h(s)` is. |
@[simp]
isSigmaCompact_preimage {s : Set Y} (h : X ≃ₜ Y) :
IsSigmaCompact (h ⁻¹' s) ↔ IsSigmaCompact s := by
rw [← image_symm]; exact h.symm.isSigmaCompact_image
@[simp] | theorem | Topology | [
"Mathlib.Logic.Equiv.Fin.Basic",
"Mathlib.Topology.Connected.LocallyConnected",
"Mathlib.Topology.DenseEmbedding",
"Mathlib.Topology.Connected.TotallyDisconnected"
] | Mathlib/Topology/Homeomorph/Lemmas.lean | isSigmaCompact_preimage | If `h : X → Y` is a homeomorphism, `h⁻¹(s)` is σ-compact iff `s` is. |
isPreconnected_image {s : Set X} (h : X ≃ₜ Y) :
IsPreconnected (h '' s) ↔ IsPreconnected s :=
⟨fun hs ↦ by simpa only [image_symm, preimage_image]
using hs.image _ h.symm.continuous.continuousOn,
fun hs ↦ hs.image _ h.continuous.continuousOn⟩
@[simp] | theorem | Topology | [
"Mathlib.Logic.Equiv.Fin.Basic",
"Mathlib.Topology.Connected.LocallyConnected",
"Mathlib.Topology.DenseEmbedding",
"Mathlib.Topology.Connected.TotallyDisconnected"
] | Mathlib/Topology/Homeomorph/Lemmas.lean | isPreconnected_image | null |
isPreconnected_preimage {s : Set Y} (h : X ≃ₜ Y) :
IsPreconnected (h ⁻¹' s) ↔ IsPreconnected s := by
rw [← image_symm, isPreconnected_image]
@[simp] | theorem | Topology | [
"Mathlib.Logic.Equiv.Fin.Basic",
"Mathlib.Topology.Connected.LocallyConnected",
"Mathlib.Topology.DenseEmbedding",
"Mathlib.Topology.Connected.TotallyDisconnected"
] | Mathlib/Topology/Homeomorph/Lemmas.lean | isPreconnected_preimage | null |
isConnected_image {s : Set X} (h : X ≃ₜ Y) :
IsConnected (h '' s) ↔ IsConnected s :=
image_nonempty.and h.isPreconnected_image
@[simp] | theorem | Topology | [
"Mathlib.Logic.Equiv.Fin.Basic",
"Mathlib.Topology.Connected.LocallyConnected",
"Mathlib.Topology.DenseEmbedding",
"Mathlib.Topology.Connected.TotallyDisconnected"
] | Mathlib/Topology/Homeomorph/Lemmas.lean | isConnected_image | null |
isConnected_preimage {s : Set Y} (h : X ≃ₜ Y) :
IsConnected (h ⁻¹' s) ↔ IsConnected s := by
rw [← image_symm, isConnected_image] | theorem | Topology | [
"Mathlib.Logic.Equiv.Fin.Basic",
"Mathlib.Topology.Connected.LocallyConnected",
"Mathlib.Topology.DenseEmbedding",
"Mathlib.Topology.Connected.TotallyDisconnected"
] | Mathlib/Topology/Homeomorph/Lemmas.lean | isConnected_preimage | null |
image_connectedComponentIn {s : Set X} (h : X ≃ₜ Y) {x : X} (hx : x ∈ s) :
h '' connectedComponentIn s x = connectedComponentIn (h '' s) (h x) := by
refine (h.continuous.image_connectedComponentIn_subset hx).antisymm ?_
have := h.symm.continuous.image_connectedComponentIn_subset (mem_image_of_mem h hx)
rwa [image_subset_iff, h.preimage_symm, h.image_symm, h.preimage_image, h.symm_apply_apply]
at this
@[simp] | theorem | Topology | [
"Mathlib.Logic.Equiv.Fin.Basic",
"Mathlib.Topology.Connected.LocallyConnected",
"Mathlib.Topology.DenseEmbedding",
"Mathlib.Topology.Connected.TotallyDisconnected"
] | Mathlib/Topology/Homeomorph/Lemmas.lean | image_connectedComponentIn | null |
comap_cocompact (h : X ≃ₜ Y) : comap h (cocompact Y) = cocompact X :=
(comap_cocompact_le h.continuous).antisymm <|
(hasBasis_cocompact.le_basis_iff (hasBasis_cocompact.comap h)).2 fun K hK =>
⟨h ⁻¹' K, h.isCompact_preimage.2 hK, Subset.rfl⟩
@[simp] | theorem | Topology | [
"Mathlib.Logic.Equiv.Fin.Basic",
"Mathlib.Topology.Connected.LocallyConnected",
"Mathlib.Topology.DenseEmbedding",
"Mathlib.Topology.Connected.TotallyDisconnected"
] | Mathlib/Topology/Homeomorph/Lemmas.lean | comap_cocompact | null |
map_cocompact (h : X ≃ₜ Y) : map h (cocompact X) = cocompact Y := by
rw [← h.comap_cocompact, map_comap_of_surjective h.surjective] | theorem | Topology | [
"Mathlib.Logic.Equiv.Fin.Basic",
"Mathlib.Topology.Connected.LocallyConnected",
"Mathlib.Topology.DenseEmbedding",
"Mathlib.Topology.Connected.TotallyDisconnected"
] | Mathlib/Topology/Homeomorph/Lemmas.lean | map_cocompact | null |
protected compactSpace [CompactSpace X] (h : X ≃ₜ Y) : CompactSpace Y where
isCompact_univ := h.symm.isCompact_preimage.2 isCompact_univ | theorem | Topology | [
"Mathlib.Logic.Equiv.Fin.Basic",
"Mathlib.Topology.Connected.LocallyConnected",
"Mathlib.Topology.DenseEmbedding",
"Mathlib.Topology.Connected.TotallyDisconnected"
] | Mathlib/Topology/Homeomorph/Lemmas.lean | compactSpace | null |
isDenseEmbedding (h : X ≃ₜ Y) : IsDenseEmbedding h :=
{ h.isEmbedding with dense := h.surjective.denseRange } | theorem | Topology | [
"Mathlib.Logic.Equiv.Fin.Basic",
"Mathlib.Topology.Connected.LocallyConnected",
"Mathlib.Topology.DenseEmbedding",
"Mathlib.Topology.Connected.TotallyDisconnected"
] | Mathlib/Topology/Homeomorph/Lemmas.lean | isDenseEmbedding | null |
protected totallyDisconnectedSpace (h : X ≃ₜ Y) [tdc : TotallyDisconnectedSpace X] :
TotallyDisconnectedSpace Y :=
(totallyDisconnectedSpace_iff Y).mpr
(h.range_coe ▸ ((IsEmbedding.isTotallyDisconnected_range h.isEmbedding).mpr tdc))
@[simp] | lemma | Topology | [
"Mathlib.Logic.Equiv.Fin.Basic",
"Mathlib.Topology.Connected.LocallyConnected",
"Mathlib.Topology.DenseEmbedding",
"Mathlib.Topology.Connected.TotallyDisconnected"
] | Mathlib/Topology/Homeomorph/Lemmas.lean | totallyDisconnectedSpace | null |
map_punctured_nhds_eq (h : X ≃ₜ Y) (x : X) : map h (𝓝[≠] x) = 𝓝[≠] (h x) := by
convert h.isEmbedding.map_nhdsWithin_eq ({x}ᶜ) x
rw [h.image_compl, Set.image_singleton]
@[simp] | theorem | Topology | [
"Mathlib.Logic.Equiv.Fin.Basic",
"Mathlib.Topology.Connected.LocallyConnected",
"Mathlib.Topology.DenseEmbedding",
"Mathlib.Topology.Connected.TotallyDisconnected"
] | Mathlib/Topology/Homeomorph/Lemmas.lean | map_punctured_nhds_eq | null |
comap_coclosedCompact (h : X ≃ₜ Y) : comap h (coclosedCompact Y) = coclosedCompact X :=
(hasBasis_coclosedCompact.comap h).eq_of_same_basis <| by
simpa [comp_def] using hasBasis_coclosedCompact.comp_surjective h.injective.preimage_surjective
@[simp] | theorem | Topology | [
"Mathlib.Logic.Equiv.Fin.Basic",
"Mathlib.Topology.Connected.LocallyConnected",
"Mathlib.Topology.DenseEmbedding",
"Mathlib.Topology.Connected.TotallyDisconnected"
] | Mathlib/Topology/Homeomorph/Lemmas.lean | comap_coclosedCompact | null |
map_coclosedCompact (h : X ≃ₜ Y) : map h (coclosedCompact X) = coclosedCompact Y := by
rw [← h.comap_coclosedCompact, map_comap_of_surjective h.surjective] | theorem | Topology | [
"Mathlib.Logic.Equiv.Fin.Basic",
"Mathlib.Topology.Connected.LocallyConnected",
"Mathlib.Topology.DenseEmbedding",
"Mathlib.Topology.Connected.TotallyDisconnected"
] | Mathlib/Topology/Homeomorph/Lemmas.lean | map_coclosedCompact | null |
locallyConnectedSpace [i : LocallyConnectedSpace Y] (h : X ≃ₜ Y) :
LocallyConnectedSpace X := by
have : ∀ x, (𝓝 x).HasBasis (fun s ↦ IsOpen s ∧ h x ∈ s ∧ IsConnected s)
(h.symm '' ·) := fun x ↦ by
rw [← h.symm_map_nhds_eq]
exact (i.1 _).map _
refine locallyConnectedSpace_of_connected_bases _ _ this fun _ _ hs ↦ ?_
exact hs.2.2.2.image _ h.symm.continuous.continuousOn | theorem | Topology | [
"Mathlib.Logic.Equiv.Fin.Basic",
"Mathlib.Topology.Connected.LocallyConnected",
"Mathlib.Topology.DenseEmbedding",
"Mathlib.Topology.Connected.TotallyDisconnected"
] | Mathlib/Topology/Homeomorph/Lemmas.lean | locallyConnectedSpace | If the codomain of a homeomorphism is a locally connected space, then the domain is also
a locally connected space. |
locallyCompactSpace_iff (h : X ≃ₜ Y) :
LocallyCompactSpace X ↔ LocallyCompactSpace Y := by
exact ⟨fun _ => h.symm.isOpenEmbedding.locallyCompactSpace,
fun _ => h.isClosedEmbedding.locallyCompactSpace⟩
@[simp] | theorem | Topology | [
"Mathlib.Logic.Equiv.Fin.Basic",
"Mathlib.Topology.Connected.LocallyConnected",
"Mathlib.Topology.DenseEmbedding",
"Mathlib.Topology.Connected.TotallyDisconnected"
] | Mathlib/Topology/Homeomorph/Lemmas.lean | locallyCompactSpace_iff | The codomain of a homeomorphism is a locally compact space if and only if
the domain is a locally compact space. |
comp_continuousOn_iff (h : X ≃ₜ Y) (f : Z → X) (s : Set Z) :
ContinuousOn (h ∘ f) s ↔ ContinuousOn f s :=
h.isInducing.continuousOn_iff.symm | theorem | Topology | [
"Mathlib.Logic.Equiv.Fin.Basic",
"Mathlib.Topology.Connected.LocallyConnected",
"Mathlib.Topology.DenseEmbedding",
"Mathlib.Topology.Connected.TotallyDisconnected"
] | Mathlib/Topology/Homeomorph/Lemmas.lean | comp_continuousOn_iff | null |
comp_continuousWithinAt_iff (h : X ≃ₜ Y) (f : Z → X) (s : Set Z) (z : Z) :
ContinuousWithinAt f s z ↔ ContinuousWithinAt (h ∘ f) s z :=
h.isInducing.continuousWithinAt_iff | theorem | Topology | [
"Mathlib.Logic.Equiv.Fin.Basic",
"Mathlib.Topology.Connected.LocallyConnected",
"Mathlib.Topology.DenseEmbedding",
"Mathlib.Topology.Connected.TotallyDisconnected"
] | Mathlib/Topology/Homeomorph/Lemmas.lean | comp_continuousWithinAt_iff | null |
@[simps!]
subtype {p : X → Prop} {q : Y → Prop} (h : X ≃ₜ Y) (h_iff : ∀ x, p x ↔ q (h x)) :
{x // p x} ≃ₜ {y // q y} where
continuous_toFun := by simpa [Equiv.coe_subtypeEquiv_eq_map] using h.continuous.subtype_map _
continuous_invFun := by simpa [Equiv.coe_subtypeEquiv_eq_map] using
h.symm.continuous.subtype_map _
__ := h.subtypeEquiv h_iff
@[simp] | def | Topology | [
"Mathlib.Logic.Equiv.Fin.Basic",
"Mathlib.Topology.Connected.LocallyConnected",
"Mathlib.Topology.DenseEmbedding",
"Mathlib.Topology.Connected.TotallyDisconnected"
] | Mathlib/Topology/Homeomorph/Lemmas.lean | subtype | A homeomorphism `h : X ≃ₜ Y` lifts to a homeomorphism between subtypes corresponding to
predicates `p : X → Prop` and `q : Y → Prop` so long as `p = q ∘ h`. |
subtype_toEquiv {p : X → Prop} {q : Y → Prop} (h : X ≃ₜ Y) (h_iff : ∀ x, p x ↔ q (h x)) :
(h.subtype h_iff).toEquiv = h.toEquiv.subtypeEquiv h_iff :=
rfl | lemma | Topology | [
"Mathlib.Logic.Equiv.Fin.Basic",
"Mathlib.Topology.Connected.LocallyConnected",
"Mathlib.Topology.DenseEmbedding",
"Mathlib.Topology.Connected.TotallyDisconnected"
] | Mathlib/Topology/Homeomorph/Lemmas.lean | subtype_toEquiv | null |
sets {s : Set X} {t : Set Y} (h : X ≃ₜ Y) (h_eq : s = h ⁻¹' t) : s ≃ₜ t :=
h.subtype <| Set.ext_iff.mp h_eq | abbrev | Topology | [
"Mathlib.Logic.Equiv.Fin.Basic",
"Mathlib.Topology.Connected.LocallyConnected",
"Mathlib.Topology.DenseEmbedding",
"Mathlib.Topology.Connected.TotallyDisconnected"
] | Mathlib/Topology/Homeomorph/Lemmas.lean | sets | A homeomorphism `h : X ≃ₜ Y` lifts to a homeomorphism between sets `s : Set X` and `t : Set Y`
whenever `h` maps `s` onto `t`. |
setCongr {s t : Set X} (h : s = t) : s ≃ₜ t where
continuous_toFun := continuous_inclusion h.subset
continuous_invFun := continuous_inclusion h.symm.subset
toEquiv := Equiv.setCongr h | def | Topology | [
"Mathlib.Logic.Equiv.Fin.Basic",
"Mathlib.Topology.Connected.LocallyConnected",
"Mathlib.Topology.DenseEmbedding",
"Mathlib.Topology.Connected.TotallyDisconnected"
] | Mathlib/Topology/Homeomorph/Lemmas.lean | setCongr | If two sets are equal, then they are homeomorphic. |
@[simps! symm_apply_snd]
prodUnique [Unique Y] :
X × Y ≃ₜ X where
toEquiv := Equiv.prodUnique X Y
continuous_toFun := continuous_fst
continuous_invFun := continuous_id.prodMk continuous_const
@[simp] theorem coe_prodUnique [Unique Y] : ⇑(prodUnique X Y) = Prod.fst := rfl | def | Topology | [
"Mathlib.Logic.Equiv.Fin.Basic",
"Mathlib.Topology.Connected.LocallyConnected",
"Mathlib.Topology.DenseEmbedding",
"Mathlib.Topology.Connected.TotallyDisconnected"
] | Mathlib/Topology/Homeomorph/Lemmas.lean | prodUnique | `X × {*}` is homeomorphic to `X`. |
@[simps! symm_apply_snd]
uniqueProd (X Y : Type*) [TopologicalSpace X] [TopologicalSpace Y] [Unique X] :
X × Y ≃ₜ Y :=
(prodComm _ _).trans (prodUnique Y X)
@[simp] theorem coe_uniqueProd [Unique X] : ⇑(uniqueProd X Y) = Prod.snd := rfl | def | Topology | [
"Mathlib.Logic.Equiv.Fin.Basic",
"Mathlib.Topology.Connected.LocallyConnected",
"Mathlib.Topology.DenseEmbedding",
"Mathlib.Topology.Connected.TotallyDisconnected"
] | Mathlib/Topology/Homeomorph/Lemmas.lean | uniqueProd | `X × {*}` is homeomorphic to `X`. |
sumPiEquivProdPi (S T : Type*) (A : S ⊕ T → Type*)
[∀ st, TopologicalSpace (A st)] :
(Π (st : S ⊕ T), A st) ≃ₜ (Π (s : S), A (.inl s)) × (Π (t : T), A (.inr t)) where
__ := Equiv.sumPiEquivProdPi _
continuous_toFun := .prodMk (by fun_prop) (by fun_prop)
continuous_invFun := continuous_pi <| by rintro (s | t) <;> dsimp <;> fun_prop | def | Topology | [
"Mathlib.Logic.Equiv.Fin.Basic",
"Mathlib.Topology.Connected.LocallyConnected",
"Mathlib.Topology.DenseEmbedding",
"Mathlib.Topology.Connected.TotallyDisconnected"
] | Mathlib/Topology/Homeomorph/Lemmas.lean | sumPiEquivProdPi | The product over `S ⊕ T` of a family of topological spaces
is homeomorphic to the product of (the product over `S`) and (the product over `T`).
This is `Equiv.sumPiEquivProdPi` as a `Homeomorph`. |
@[simps! -fullyApplied]
piUnique {α : Type*} [Unique α] (f : α → Type*) [∀ x, TopologicalSpace (f x)] :
(Π t, f t) ≃ₜ f default :=
(Equiv.piUnique f).toHomeomorphOfContinuousOpen (continuous_apply default) (isOpenMap_eval _) | def | Topology | [
"Mathlib.Logic.Equiv.Fin.Basic",
"Mathlib.Topology.Connected.LocallyConnected",
"Mathlib.Topology.DenseEmbedding",
"Mathlib.Topology.Connected.TotallyDisconnected"
] | Mathlib/Topology/Homeomorph/Lemmas.lean | piUnique | The product `Π t : α, f t` of a family of topological spaces is homeomorphic to the
space `f ⬝` when `α` only contains `⬝`.
This is `Equiv.piUnique` as a `Homeomorph`. |
@[simps +simpRhs toEquiv, simps! -isSimp apply]
piCongrLeft {ι ι' : Type*} {Y : ι' → Type*} [∀ j, TopologicalSpace (Y j)]
(e : ι ≃ ι') : (∀ i, Y (e i)) ≃ₜ ∀ j, Y j where
continuous_toFun := continuous_pi <| e.forall_congr_right.mp fun i ↦ by
simpa only [Equiv.toFun_as_coe, Equiv.piCongrLeft_apply_apply] using continuous_apply i
continuous_invFun := Pi.continuous_precomp' e
toEquiv := Equiv.piCongrLeft _ e
@[simp] | def | Topology | [
"Mathlib.Logic.Equiv.Fin.Basic",
"Mathlib.Topology.Connected.LocallyConnected",
"Mathlib.Topology.DenseEmbedding",
"Mathlib.Topology.Connected.TotallyDisconnected"
] | Mathlib/Topology/Homeomorph/Lemmas.lean | piCongrLeft | `Equiv.piCongrLeft` as a homeomorphism: this is the natural homeomorphism
`Π i, Y (e i) ≃ₜ Π j, Y j` obtained from a bijection `ι ≃ ι'`. |
piCongrLeft_refl {ι : Type*} {X : ι → Type*} [∀ i, TopologicalSpace (X i)] :
piCongrLeft (.refl ι) = .refl (∀ i, X i) :=
rfl
@[simp] | lemma | Topology | [
"Mathlib.Logic.Equiv.Fin.Basic",
"Mathlib.Topology.Connected.LocallyConnected",
"Mathlib.Topology.DenseEmbedding",
"Mathlib.Topology.Connected.TotallyDisconnected"
] | Mathlib/Topology/Homeomorph/Lemmas.lean | piCongrLeft_refl | null |
piCongrLeft_symm_apply {ι ι' : Type*} {Y : ι' → Type*} [∀ j, TopologicalSpace (Y j)]
(e : ι ≃ ι') : ⇑(piCongrLeft (Y := Y) e).symm = (· <| e ·) :=
rfl
@[simp] | lemma | Topology | [
"Mathlib.Logic.Equiv.Fin.Basic",
"Mathlib.Topology.Connected.LocallyConnected",
"Mathlib.Topology.DenseEmbedding",
"Mathlib.Topology.Connected.TotallyDisconnected"
] | Mathlib/Topology/Homeomorph/Lemmas.lean | piCongrLeft_symm_apply | null |
piCongrLeft_apply_apply {ι ι' : Type*} {Y : ι' → Type*} [∀ j, TopologicalSpace (Y j)]
(e : ι ≃ ι') (x : ∀ i, Y (e i)) (i : ι) : piCongrLeft e x (e i) = x i :=
Equiv.piCongrLeft_apply_apply .. | lemma | Topology | [
"Mathlib.Logic.Equiv.Fin.Basic",
"Mathlib.Topology.Connected.LocallyConnected",
"Mathlib.Topology.DenseEmbedding",
"Mathlib.Topology.Connected.TotallyDisconnected"
] | Mathlib/Topology/Homeomorph/Lemmas.lean | piCongrLeft_apply_apply | null |
@[simps! apply toEquiv]
piCongrRight {ι : Type*} {Y₁ Y₂ : ι → Type*} [∀ i, TopologicalSpace (Y₁ i)]
[∀ i, TopologicalSpace (Y₂ i)] (F : ∀ i, Y₁ i ≃ₜ Y₂ i) : (∀ i, Y₁ i) ≃ₜ ∀ i, Y₂ i where
continuous_toFun := Pi.continuous_postcomp' fun i ↦ (F i).continuous
continuous_invFun := Pi.continuous_postcomp' fun i ↦ (F i).symm.continuous
toEquiv := Equiv.piCongrRight fun i => (F i).toEquiv
@[simp] | def | Topology | [
"Mathlib.Logic.Equiv.Fin.Basic",
"Mathlib.Topology.Connected.LocallyConnected",
"Mathlib.Topology.DenseEmbedding",
"Mathlib.Topology.Connected.TotallyDisconnected"
] | Mathlib/Topology/Homeomorph/Lemmas.lean | piCongrRight | `Equiv.piCongrRight` as a homeomorphism: this is the natural homeomorphism
`Π i, Y₁ i ≃ₜ Π j, Y₂ i` obtained from homeomorphisms `Y₁ i ≃ₜ Y₂ i` for each `i`. |
piCongrRight_symm {ι : Type*} {Y₁ Y₂ : ι → Type*} [∀ i, TopologicalSpace (Y₁ i)]
[∀ i, TopologicalSpace (Y₂ i)] (F : ∀ i, Y₁ i ≃ₜ Y₂ i) :
(piCongrRight F).symm = piCongrRight fun i => (F i).symm :=
rfl | theorem | Topology | [
"Mathlib.Logic.Equiv.Fin.Basic",
"Mathlib.Topology.Connected.LocallyConnected",
"Mathlib.Topology.DenseEmbedding",
"Mathlib.Topology.Connected.TotallyDisconnected"
] | Mathlib/Topology/Homeomorph/Lemmas.lean | piCongrRight_symm | null |
@[simps! apply toEquiv]
piCongr {ι₁ ι₂ : Type*} {Y₁ : ι₁ → Type*} {Y₂ : ι₂ → Type*}
[∀ i₁, TopologicalSpace (Y₁ i₁)] [∀ i₂, TopologicalSpace (Y₂ i₂)]
(e : ι₁ ≃ ι₂) (F : ∀ i₁, Y₁ i₁ ≃ₜ Y₂ (e i₁)) : (∀ i₁, Y₁ i₁) ≃ₜ ∀ i₂, Y₂ i₂ :=
(Homeomorph.piCongrRight F).trans (Homeomorph.piCongrLeft e) | def | Topology | [
"Mathlib.Logic.Equiv.Fin.Basic",
"Mathlib.Topology.Connected.LocallyConnected",
"Mathlib.Topology.DenseEmbedding",
"Mathlib.Topology.Connected.TotallyDisconnected"
] | Mathlib/Topology/Homeomorph/Lemmas.lean | piCongr | `Equiv.piCongr` as a homeomorphism: this is the natural homeomorphism
`Π i₁, Y₁ i ≃ₜ Π i₂, Y₂ i₂` obtained from a bijection `ι₁ ≃ ι₂` and homeomorphisms
`Y₁ i₁ ≃ₜ Y₂ (e i₁)` for each `i₁ : ι₁`. |
ulift.{u, v} {X : Type v} [TopologicalSpace X] : ULift.{u, v} X ≃ₜ X where
continuous_toFun := continuous_uliftDown
continuous_invFun := continuous_uliftUp
toEquiv := Equiv.ulift | def | Topology | [
"Mathlib.Logic.Equiv.Fin.Basic",
"Mathlib.Topology.Connected.LocallyConnected",
"Mathlib.Topology.DenseEmbedding",
"Mathlib.Topology.Connected.TotallyDisconnected"
] | Mathlib/Topology/Homeomorph/Lemmas.lean | ulift. | `ULift X` is homeomorphic to `X`. |
@[simps!]
sumArrowHomeomorphProdArrow {ι ι' : Type*} : (ι ⊕ ι' → X) ≃ₜ (ι → X) × (ι' → X) where
toEquiv := Equiv.sumArrowEquivProdArrow _ _ _
continuous_toFun := by
dsimp [Equiv.sumArrowEquivProdArrow]
fun_prop
continuous_invFun := continuous_pi fun i ↦ match i with
| .inl i => by apply (continuous_apply _).comp' continuous_fst
| .inr i => by apply (continuous_apply _).comp' 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 | sumArrowHomeomorphProdArrow | The natural homeomorphism `(ι ⊕ ι' → X) ≃ₜ (ι → X) × (ι' → X)`.
`Equiv.sumArrowEquivProdArrow` as a homeomorphism. |
private _root_.Fin.appendEquiv_eq_homeomorph (m n : ℕ) : Fin.appendEquiv m n =
((sumArrowHomeomorphProdArrow).symm.trans
(piCongrLeft (Y := fun _ ↦ X) finSumFinEquiv)).toEquiv := by
apply Equiv.symm_bijective.injective
ext x i <;> simp | 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.appendEquiv_eq_homeomorph | null |
_root_.Fin.continuous_append (m n : ℕ) :
Continuous fun (p : (Fin m → X) × (Fin n → X)) ↦ Fin.append p.1 p.2 := by
suffices Continuous (Fin.appendEquiv m n) by exact this
rw [Fin.appendEquiv_eq_homeomorph]
exact Homeomorph.continuous_toFun _ | 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.continuous_append | null |
@[simps!]
_root_.Fin.appendHomeomorph (m n : ℕ) : (Fin m → X) × (Fin n → X) ≃ₜ (Fin (m + n) → X) where
toEquiv := Fin.appendEquiv m n
continuous_toFun := Fin.continuous_append m n
continuous_invFun := by
rw [Fin.appendEquiv_eq_homeomorph]
exact Homeomorph.continuous_invFun _
@[simp] | def | 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 | The natural homeomorphism between `(Fin m → X) × (Fin n → X)` and `Fin (m + n) → X`.
`Fin.appendEquiv` as a homeomorphism |
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