Partial homeomorphisms

Partial homeomorphisms

This file defines homeomorphisms between open subsets of topological spaces. An element e of PartialHomeomorph α β is an extension of PartialEquiv α β, i.e., it is a pair of functions e.toFun and e.invFun, inverse of each other on the sets e.source and e.target. Additionally, we require that these sets are open, and that the functions are continuous on them. Equivalently, they are homeomorphisms there.

As in equivs, we register a coercion to functions, and we use e x and e.symm x throughout instead of e.toFun x and e.invFun x.

Main definitions

• Homeomorph.toPartialHomeomorph: associating a partial homeomorphism to a homeomorphism, with source = target = Set.univ;
• PartialHomeomorph.symm: the inverse of a partial homeomorphism
• PartialHomeomorph.trans: the composition of two partial homeomorphisms
• PartialHomeomorph.refl: the identity partial homeomorphism
• PartialHomeomorph.ofSet: the identity on a set s
• PartialHomeomorph.EqOnSource: equivalence relation describing the "right" notion of equality for partial homeomorphisms

Implementation notes

Most statements are copied from their PartialEquiv versions, although some care is required especially when restricting to subsets, as these should be open subsets.

For design notes, see PartialEquiv.lean.

Local coding conventions

If a lemma deals with the intersection of a set with either source or target of a PartialEquiv, then it should use e.source ∩ s or e.target ∩ t, not s ∩ e.source or t ∩ e.target.

structure PartialHomeomorph (α : Type u_5) (β : Type u_6) [TopologicalSpace α] [TopologicalSpace β] extends PartialEquiv : Type (max u_5 u_6)

Partial homeomorphisms, defined on open subsets of the space

• toFun : α → β
• invFun : β → α
• source : Set α
• target : Set β
• map_source' : ∀ ⦃x : α⦄, x ∈ s.source → ↑s.toPartialEquiv x ∈ s.target
• map_target' : ∀ ⦃x : β⦄, x ∈ s.target → PartialEquiv.invFun s.toPartialEquiv x ∈ s.source
• left_inv' : ∀ ⦃x : α⦄, x ∈ s.source → PartialEquiv.invFun s.toPartialEquiv (↑s.toPartialEquiv x) = x
• right_inv' : ∀ ⦃x : β⦄, x ∈ s.target → ↑s.toPartialEquiv (PartialEquiv.invFun s.toPartialEquiv x) = x
• open_source : IsOpen s.source
• open_target : IsOpen s.target
• continuousOn_toFun : ContinuousOn (↑s.toPartialEquiv) s.source
• continuousOn_invFun : ContinuousOn s.invFun s.target

def PartialHomeomorph.toFun' {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] (e : PartialHomeomorph α β) : α → β

Coercion of a partial homeomorphisms to a function. We don't use e.toFun because it is actually e.toPartialEquiv.toFun, so simp will apply lemmas about toPartialEquiv. While we may want to switch to this behavior later, doing it mid-port will break a lot of proofs.

Equations

• ↑e = ↑e.toPartialEquiv

instance PartialHomeomorph.instCoeFunPartialHomeomorphForAll {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] : CoeFun (PartialHomeomorph α β) fun (x : PartialHomeomorph α β) => α → β

Coercion of a PartialHomeomorph to function. Note that a PartialHomeomorph is not FunLike.

Equations

• PartialHomeomorph.instCoeFunPartialHomeomorphForAll = { coe := fun (e : PartialHomeomorph α β) => ↑e }

def PartialHomeomorph.symm {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] (e : PartialHomeomorph α β) : PartialHomeomorph β α

The inverse of a partial homeomorphism

Equations

• One or more equations did not get rendered due to their size.

def PartialHomeomorph.Simps.apply {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] (e : PartialHomeomorph α β) : α → β

See Note [custom simps projection]. We need to specify this projection explicitly in this case, because it is a composition of multiple projections.

Equations

• PartialHomeomorph.Simps.apply e = ↑e

def PartialHomeomorph.Simps.symm_apply {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] (e : PartialHomeomorph α β) : β → α

See Note [custom simps projection]

Equations

• PartialHomeomorph.Simps.symm_apply e = ↑(PartialHomeomorph.symm e)

theorem PartialHomeomorph.continuousOn {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] (e : PartialHomeomorph α β) : ContinuousOn (↑e) e.source

theorem PartialHomeomorph.continuousOn_symm {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] (e : PartialHomeomorph α β) : ContinuousOn (↑(PartialHomeomorph.symm e)) e.target

@[simp]

theorem PartialHomeomorph.mk_coe {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] (e : PartialEquiv α β) (a : IsOpen e.source) (b : IsOpen e.target) (c : ContinuousOn (↑e) e.source) (d : ContinuousOn e.invFun e.target) : ↑{ toPartialEquiv := e, open_source := a, open_target := b, continuousOn_toFun := c, continuousOn_invFun := d } = ↑e

@[simp]

theorem PartialHomeomorph.mk_coe_symm {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] (e : PartialEquiv α β) (a : IsOpen e.source) (b : IsOpen e.target) (c : ContinuousOn (↑e) e.source) (d : ContinuousOn e.invFun e.target) : ↑(PartialHomeomorph.symm { toPartialEquiv := e, open_source := a, open_target := b, continuousOn_toFun := c, continuousOn_invFun := d }) = ↑(PartialEquiv.symm e)

theorem PartialHomeomorph.toPartialEquiv_injective {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] : Function.Injective PartialHomeomorph.toPartialEquiv

@[simp]

theorem PartialHomeomorph.toFun_eq_coe {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] (e : PartialHomeomorph α β) : ↑e.toPartialEquiv = ↑e

@[simp]

theorem PartialHomeomorph.invFun_eq_coe {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] (e : PartialHomeomorph α β) : e.invFun = ↑(PartialHomeomorph.symm e)

@[simp]

theorem PartialHomeomorph.coe_coe {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] (e : PartialHomeomorph α β) : ↑e.toPartialEquiv = ↑e

@[simp]

theorem PartialHomeomorph.coe_coe_symm {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] (e : PartialHomeomorph α β) : ↑(PartialEquiv.symm e.toPartialEquiv) = ↑(PartialHomeomorph.symm e)

@[simp]

theorem PartialHomeomorph.map_source {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] (e : PartialHomeomorph α β) {x : α} (h : x ∈ e.source) : ↑e x ∈ e.target

theorem PartialHomeomorph.map_source'' {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] (e : PartialHomeomorph α β) : ↑e '' e.source ⊆ e.target

Variant of map_source, stated for images of subsets of source.

@[simp]

theorem PartialHomeomorph.map_target {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] (e : PartialHomeomorph α β) {x : β} (h : x ∈ e.target) : ↑(PartialHomeomorph.symm e) x ∈ e.source

@[simp]

theorem PartialHomeomorph.left_inv {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] (e : PartialHomeomorph α β) {x : α} (h : x ∈ e.source) : ↑(PartialHomeomorph.symm e) (↑e x) = x

@[simp]

theorem PartialHomeomorph.right_inv {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] (e : PartialHomeomorph α β) {x : β} (h : x ∈ e.target) : ↑e (↑(PartialHomeomorph.symm e) x) = x

theorem PartialHomeomorph.eq_symm_apply {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] (e : PartialHomeomorph α β) {x : α} {y : β} (hx : x ∈ e.source) (hy : y ∈ e.target) : x = ↑(PartialHomeomorph.symm e) y ↔ ↑e x = y

theorem PartialHomeomorph.mapsTo {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] (e : PartialHomeomorph α β) : Set.MapsTo (↑e) e.source e.target

theorem PartialHomeomorph.symm_mapsTo {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] (e : PartialHomeomorph α β) : Set.MapsTo (↑(PartialHomeomorph.symm e)) e.target e.source

theorem PartialHomeomorph.leftInvOn {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] (e : PartialHomeomorph α β) : Set.LeftInvOn (↑(PartialHomeomorph.symm e)) (↑e) e.source

theorem PartialHomeomorph.rightInvOn {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] (e : PartialHomeomorph α β) : Set.RightInvOn (↑(PartialHomeomorph.symm e)) (↑e) e.target

theorem PartialHomeomorph.invOn {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] (e : PartialHomeomorph α β) : Set.InvOn (↑(PartialHomeomorph.symm e)) (↑e) e.source e.target

theorem PartialHomeomorph.injOn {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] (e : PartialHomeomorph α β) : Set.InjOn (↑e) e.source

theorem PartialHomeomorph.bijOn {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] (e : PartialHomeomorph α β) : Set.BijOn (↑e) e.source e.target

theorem PartialHomeomorph.surjOn {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] (e : PartialHomeomorph α β) : Set.SurjOn (↑e) e.source e.target

@[simp]

theorem Homeomorph.toPartialHomeomorphOfImageEq_toPartialEquiv {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] (e : α ≃ₜ β) (s : Set α) (hs : IsOpen s) (t : Set β) (h : ⇑e '' s = t) : (Homeomorph.toPartialHomeomorphOfImageEq e s hs t h).toPartialEquiv = Equiv.toPartialEquivOfImageEq e.toEquiv s t h

@[simp]

theorem Homeomorph.toPartialHomeomorphOfImageEq_symm_apply {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] (e : α ≃ₜ β) (s : Set α) (hs : IsOpen s) (t : Set β) (h : ⇑e '' s = t) : ↑(PartialHomeomorph.symm (Homeomorph.toPartialHomeomorphOfImageEq e s hs t h)) = ⇑(Homeomorph.symm e)

@[simp]

theorem Homeomorph.toPartialHomeomorphOfImageEq_apply {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] (e : α ≃ₜ β) (s : Set α) (hs : IsOpen s) (t : Set β) (h : ⇑e '' s = t) : ↑(Homeomorph.toPartialHomeomorphOfImageEq e s hs t h) = ⇑e

theorem Homeomorph.toPartialHomeomorphOfImageEq_source {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] (e : α ≃ₜ β) (s : Set α) (hs : IsOpen s) (t : Set β) (h : ⇑e '' s = t) : (Homeomorph.toPartialHomeomorphOfImageEq e s hs t h).toPartialEquiv.source = s

theorem Homeomorph.toPartialHomeomorphOfImageEq_target {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] (e : α ≃ₜ β) (s : Set α) (hs : IsOpen s) (t : Set β) (h : ⇑e '' s = t) : (Homeomorph.toPartialHomeomorphOfImageEq e s hs t h).toPartialEquiv.target = t

def Homeomorph.toPartialHomeomorphOfImageEq {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] (e : α ≃ₜ β) (s : Set α) (hs : IsOpen s) (t : Set β) (h : ⇑e '' s = t) : PartialHomeomorph α β

Interpret a Homeomorph as a PartialHomeomorph by restricting it to an open set s in the domain and to t in the codomain.

Equations

• One or more equations did not get rendered due to their size.

@[simp]

theorem Homeomorph.toPartialHomeomorph_symm_apply {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] (e : α ≃ₜ β) : ↑(PartialHomeomorph.symm (Homeomorph.toPartialHomeomorph e)) = ⇑(Homeomorph.symm e)

@[simp]

theorem Homeomorph.toPartialHomeomorph_apply {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] (e : α ≃ₜ β) : ↑(Homeomorph.toPartialHomeomorph e) = ⇑e

@[simp]

theorem Homeomorph.toPartialHomeomorph_source {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] (e : α ≃ₜ β) : (Homeomorph.toPartialHomeomorph e).toPartialEquiv.source = Set.univ

@[simp]

theorem Homeomorph.toPartialHomeomorph_target {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] (e : α ≃ₜ β) : (Homeomorph.toPartialHomeomorph e).toPartialEquiv.target = Set.univ

def Homeomorph.toPartialHomeomorph {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] (e : α ≃ₜ β) : PartialHomeomorph α β

A homeomorphism induces a partial homeomorphism on the whole space

Equations

• Homeomorph.toPartialHomeomorph e = Homeomorph.toPartialHomeomorphOfImageEq e Set.univ (_ : IsOpen Set.univ) Set.univ (_ : ⇑e '' Set.univ = Set.univ)

def PartialHomeomorph.replaceEquiv {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] (e : PartialHomeomorph α β) (e' : PartialEquiv α β) (h : e.toPartialEquiv = e') : PartialHomeomorph α β

Replace toPartialEquiv field to provide better definitional equalities.

Equations

• One or more equations did not get rendered due to their size.

theorem PartialHomeomorph.replaceEquiv_eq_self {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] (e : PartialHomeomorph α β) (e' : PartialEquiv α β) (h : e.toPartialEquiv = e') : PartialHomeomorph.replaceEquiv e e' h = e

theorem PartialHomeomorph.source_preimage_target {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] (e : PartialHomeomorph α β) : e.source ⊆ ↑e ⁻¹' e.target

@[deprecated PartialHomeomorph.toPartialEquiv_injective]

theorem PartialHomeomorph.eq_of_localEquiv_eq {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] {e : PartialHomeomorph α β} {e' : PartialHomeomorph α β} (h : e.toPartialEquiv = e'.toPartialEquiv) : e = e'

theorem PartialHomeomorph.eventually_left_inverse {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] (e : PartialHomeomorph α β) {x : α} (hx : x ∈ e.source) : ∀ᶠ (y : α) in nhds x, ↑(PartialHomeomorph.symm e) (↑e y) = y

theorem PartialHomeomorph.eventually_left_inverse' {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] (e : PartialHomeomorph α β) {x : β} (hx : x ∈ e.target) : ∀ᶠ (y : α) in nhds (↑(PartialHomeomorph.symm e) x), ↑(PartialHomeomorph.symm e) (↑e y) = y

theorem PartialHomeomorph.eventually_right_inverse {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] (e : PartialHomeomorph α β) {x : β} (hx : x ∈ e.target) : ∀ᶠ (y : β) in nhds x, ↑e (↑(PartialHomeomorph.symm e) y) = y

theorem PartialHomeomorph.eventually_right_inverse' {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] (e : PartialHomeomorph α β) {x : α} (hx : x ∈ e.source) : ∀ᶠ (y : β) in nhds (↑e x), ↑e (↑(PartialHomeomorph.symm e) y) = y

theorem PartialHomeomorph.eventually_ne_nhdsWithin {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] (e : PartialHomeomorph α β) {x : α} (hx : x ∈ e.source) : ∀ᶠ (x' : α) in nhdsWithin x {x}ᶜ, ↑e x' ≠ ↑e x

theorem PartialHomeomorph.nhdsWithin_source_inter {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] (e : PartialHomeomorph α β) {x : α} (hx : x ∈ e.source) (s : Set α) : nhdsWithin x (e.source ∩ s) = nhdsWithin x s

theorem PartialHomeomorph.nhdsWithin_target_inter {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] (e : PartialHomeomorph α β) {x : β} (hx : x ∈ e.target) (s : Set β) : nhdsWithin x (e.target ∩ s) = nhdsWithin x s

theorem PartialHomeomorph.image_eq_target_inter_inv_preimage {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] (e : PartialHomeomorph α β) {s : Set α} (h : s ⊆ e.source) : ↑e '' s = e.target ∩ ↑(PartialHomeomorph.symm e) ⁻¹' s

theorem PartialHomeomorph.image_source_inter_eq' {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] (e : PartialHomeomorph α β) (s : Set α) : ↑e '' (e.source ∩ s) = e.target ∩ ↑(PartialHomeomorph.symm e) ⁻¹' s

theorem PartialHomeomorph.image_source_inter_eq {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] (e : PartialHomeomorph α β) (s : Set α) : ↑e '' (e.source ∩ s) = e.target ∩ ↑(PartialHomeomorph.symm e) ⁻¹' (e.source ∩ s)

theorem PartialHomeomorph.symm_image_eq_source_inter_preimage {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] (e : PartialHomeomorph α β) {s : Set β} (h : s ⊆ e.target) : ↑(PartialHomeomorph.symm e) '' s = e.source ∩ ↑e ⁻¹' s

theorem PartialHomeomorph.symm_image_target_inter_eq {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] (e : PartialHomeomorph α β) (s : Set β) : ↑(PartialHomeomorph.symm e) '' (e.target ∩ s) = e.source ∩ ↑e ⁻¹' (e.target ∩ s)

theorem PartialHomeomorph.source_inter_preimage_inv_preimage {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] (e : PartialHomeomorph α β) (s : Set α) : e.source ∩ ↑e ⁻¹' (↑(PartialHomeomorph.symm e) ⁻¹' s) = e.source ∩ s

theorem PartialHomeomorph.target_inter_inv_preimage_preimage {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] (e : PartialHomeomorph α β) (s : Set β) : e.target ∩ ↑(PartialHomeomorph.symm e) ⁻¹' (↑e ⁻¹' s) = e.target ∩ s

theorem PartialHomeomorph.source_inter_preimage_target_inter {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] (e : PartialHomeomorph α β) (s : Set β) : e.source ∩ ↑e ⁻¹' (e.target ∩ s) = e.source ∩ ↑e ⁻¹' s

theorem PartialHomeomorph.image_source_eq_target {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] (e : PartialHomeomorph α β) : ↑e '' e.source = e.target

theorem PartialHomeomorph.symm_image_target_eq_source {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] (e : PartialHomeomorph α β) : ↑(PartialHomeomorph.symm e) '' e.target = e.source

theorem PartialHomeomorph.ext {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] (e : PartialHomeomorph α β) (e' : PartialHomeomorph α β) (h : ∀ (x : α), ↑e x = ↑e' x) (hinv : ∀ (x : β), ↑(PartialHomeomorph.symm e) x = ↑(PartialHomeomorph.symm e') x) (hs : e.source = e'.source) : e = e'

Two partial homeomorphisms are equal when they have equal toFun, invFun and source. It is not sufficient to have equal toFun and source, as this only determines invFun on the target. This would only be true for a weaker notion of equality, arguably the right one, called EqOnSource.

theorem PartialHomeomorph.ext_iff {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] {e : PartialHomeomorph α β} {e' : PartialHomeomorph α β} : e = e' ↔ (∀ (x : α), ↑e x = ↑e' x) ∧ (∀ (x : β), ↑(PartialHomeomorph.symm e) x = ↑(PartialHomeomorph.symm e') x) ∧ e.source = e'.source

@[simp]

theorem PartialHomeomorph.symm_toPartialEquiv {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] (e : PartialHomeomorph α β) : (PartialHomeomorph.symm e).toPartialEquiv = PartialEquiv.symm e.toPartialEquiv

theorem PartialHomeomorph.symm_source {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] (e : PartialHomeomorph α β) : (PartialHomeomorph.symm e).toPartialEquiv.source = e.target

theorem PartialHomeomorph.symm_target {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] (e : PartialHomeomorph α β) : (PartialHomeomorph.symm e).toPartialEquiv.target = e.source

@[simp]

theorem PartialHomeomorph.symm_symm {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] (e : PartialHomeomorph α β) : PartialHomeomorph.symm (PartialHomeomorph.symm e) = e

theorem PartialHomeomorph.symm_bijective {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] : Function.Bijective PartialHomeomorph.symm

theorem PartialHomeomorph.continuousAt {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] (e : PartialHomeomorph α β) {x : α} (h : x ∈ e.source) : ContinuousAt (↑e) x

A partial homeomorphism is continuous at any point of its source

theorem PartialHomeomorph.continuousAt_symm {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] (e : PartialHomeomorph α β) {x : β} (h : x ∈ e.target) : ContinuousAt (↑(PartialHomeomorph.symm e)) x

A partial homeomorphism inverse is continuous at any point of its target

theorem PartialHomeomorph.tendsto_symm {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] (e : PartialHomeomorph α β) {x : α} (hx : x ∈ e.source) : Filter.Tendsto (↑(PartialHomeomorph.symm e)) (nhds (↑e x)) (nhds x)

theorem PartialHomeomorph.map_nhds_eq {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] (e : PartialHomeomorph α β) {x : α} (hx : x ∈ e.source) : Filter.map (↑e) (nhds x) = nhds (↑e x)

theorem PartialHomeomorph.symm_map_nhds_eq {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] (e : PartialHomeomorph α β) {x : α} (hx : x ∈ e.source) : Filter.map (↑(PartialHomeomorph.symm e)) (nhds (↑e x)) = nhds x

theorem PartialHomeomorph.image_mem_nhds {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] (e : PartialHomeomorph α β) {x : α} (hx : x ∈ e.source) {s : Set α} (hs : s ∈ nhds x) : ↑e '' s ∈ nhds (↑e x)

theorem PartialHomeomorph.map_nhdsWithin_eq {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] (e : PartialHomeomorph α β) {x : α} (hx : x ∈ e.source) (s : Set α) : Filter.map (↑e) (nhdsWithin x s) = nhdsWithin (↑e x) (↑e '' (e.source ∩ s))

theorem PartialHomeomorph.map_nhdsWithin_preimage_eq {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] (e : PartialHomeomorph α β) {x : α} (hx : x ∈ e.source) (s : Set β) : Filter.map (↑e) (nhdsWithin x (↑e ⁻¹' s)) = nhdsWithin (↑e x) s

theorem PartialHomeomorph.eventually_nhds {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] (e : PartialHomeomorph α β) {x : α} (p : β → Prop) (hx : x ∈ e.source) : (∀ᶠ (y : β) in nhds (↑e x), p y) ↔ ∀ᶠ (x : α) in nhds x, p (↑e x)

theorem PartialHomeomorph.eventually_nhds' {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] (e : PartialHomeomorph α β) {x : α} (p : α → Prop) (hx : x ∈ e.source) : (∀ᶠ (y : β) in nhds (↑e x), p (↑(PartialHomeomorph.symm e) y)) ↔ ∀ᶠ (x : α) in nhds x, p x

theorem PartialHomeomorph.eventually_nhdsWithin {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] (e : PartialHomeomorph α β) {x : α} (p : β → Prop) {s : Set α} (hx : x ∈ e.source) : (∀ᶠ (y : β) in nhdsWithin (↑e x) (↑(PartialHomeomorph.symm e) ⁻¹' s), p y) ↔ ∀ᶠ (x : α) in nhdsWithin x s, p (↑e x)

theorem PartialHomeomorph.eventually_nhdsWithin' {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] (e : PartialHomeomorph α β) {x : α} (p : α → Prop) {s : Set α} (hx : x ∈ e.source) : (∀ᶠ (y : β) in nhdsWithin (↑e x) (↑(PartialHomeomorph.symm e) ⁻¹' s), p (↑(PartialHomeomorph.symm e) y)) ↔ ∀ᶠ (x : α) in nhdsWithin x s, p x

theorem PartialHomeomorph.preimage_eventuallyEq_target_inter_preimage_inter {α : Type u_1} {β : Type u_2} {γ : Type u_3} [TopologicalSpace α] [TopologicalSpace β] [TopologicalSpace γ] {e : PartialHomeomorph α β} {s : Set α} {t : Set γ} {x : α} {f : α → γ} (hf : ContinuousWithinAt f s x) (hxe : x ∈ e.source) (ht : t ∈ nhds (f x)) : ↑(PartialHomeomorph.symm e) ⁻¹' s =ᶠ[nhds (↑e x)] e.target ∩ ↑(PartialHomeomorph.symm e) ⁻¹' (s ∩ f ⁻¹' t)

This lemma is useful in the manifold library in the case that e is a chart. It states that locally around e x the set e.symm ⁻¹' s is the same as the set intersected with the target of e and some other neighborhood of f x (which will be the source of a chart on γ).

theorem PartialHomeomorph.isOpen_inter_preimage {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] (e : PartialHomeomorph α β) {s : Set β} (hs : IsOpen s) : IsOpen (e.source ∩ ↑e ⁻¹' s)

theorem PartialHomeomorph.isOpen_inter_preimage_symm {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] (e : PartialHomeomorph α β) {s : Set α} (hs : IsOpen s) : IsOpen (e.target ∩ ↑(PartialHomeomorph.symm e) ⁻¹' s)

theorem PartialHomeomorph.isOpen_image_of_subset_source {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] (e : PartialHomeomorph α β) {s : Set α} (hs : IsOpen s) (hse : s ⊆ e.source) : IsOpen (↑e '' s)

A partial homeomorphism is an open map on its source: the image of an open subset of the source is open.

theorem PartialHomeomorph.isOpen_image_source_inter {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] (e : PartialHomeomorph α β) {s : Set α} (hs : IsOpen s) : IsOpen (↑e '' (e.source ∩ s))

The image of the restriction of an open set to the source is open.

theorem PartialHomeomorph.isOpen_image_symm_of_subset_target {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] (e : PartialHomeomorph α β) {t : Set β} (ht : IsOpen t) (hte : t ⊆ e.target) : IsOpen (↑(PartialHomeomorph.symm e) '' t)

The inverse of a partial homeomorphism e is an open map on e.target.

theorem PartialHomeomorph.isOpen_symm_image_iff_of_subset_target {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] (e : PartialHomeomorph α β) {t : Set β} (hs : t ⊆ e.target) : IsOpen (↑(PartialHomeomorph.symm e) '' t) ↔ IsOpen t

theorem PartialHomeomorph.isOpen_image_iff_of_subset_source {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] (e : PartialHomeomorph α β) {s : Set α} (hs : s ⊆ e.source) : IsOpen (↑e '' s) ↔ IsOpen s

PartialHomeomorph.IsImage relation

We say that t : Set β is an image of s : Set α under a partial homeomorphism e if any of the following equivalent conditions hold:

• e '' (e.source ∩ s) = e.target ∩ t;
• e.source ∩ e ⁻¹ t = e.source ∩ s;
• ∀ x ∈ e.source, e x ∈ t ↔ x ∈ s (this one is used in the definition).

This definition is a restatement of PartialEquiv.IsImage for partial homeomorphisms. In this section we transfer API about PartialEquiv.IsImage to partial homeomorphisms and add a few PartialHomeomorph-specific lemmas like PartialHomeomorph.IsImage.closure.

def PartialHomeomorph.IsImage {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] (e : PartialHomeomorph α β) (s : Set α) (t : Set β) : Prop

We say that t : Set β is an image of s : Set α under a partial homeomorphism e if any of the following equivalent conditions hold:

• e '' (e.source ∩ s) = e.target ∩ t;
• e.source ∩ e ⁻¹ t = e.source ∩ s;
• ∀ x ∈ e.source, e x ∈ t ↔ x ∈ s (this one is used in the definition).

Equations

• PartialHomeomorph.IsImage e s t = ∀ ⦃x : α⦄, x ∈ e.source → (↑e x ∈ t ↔ x ∈ s)

theorem PartialHomeomorph.IsImage.toPartialEquiv {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] {e : PartialHomeomorph α β} {s : Set α} {t : Set β} (h : PartialHomeomorph.IsImage e s t) : PartialEquiv.IsImage e.toPartialEquiv s t

theorem PartialHomeomorph.IsImage.apply_mem_iff {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] {e : PartialHomeomorph α β} {s : Set α} {t : Set β} {x : α} (h : PartialHomeomorph.IsImage e s t) (hx : x ∈ e.source) : ↑e x ∈ t ↔ x ∈ s

theorem PartialHomeomorph.IsImage.symm {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] {e : PartialHomeomorph α β} {s : Set α} {t : Set β} (h : PartialHomeomorph.IsImage e s t) : PartialHomeomorph.IsImage (PartialHomeomorph.symm e) t s

theorem PartialHomeomorph.IsImage.symm_apply_mem_iff {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] {e : PartialHomeomorph α β} {s : Set α} {t : Set β} {y : β} (h : PartialHomeomorph.IsImage e s t) (hy : y ∈ e.target) : ↑(PartialHomeomorph.symm e) y ∈ s ↔ y ∈ t

@[simp]

theorem PartialHomeomorph.IsImage.symm_iff {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] {e : PartialHomeomorph α β} {s : Set α} {t : Set β} : PartialHomeomorph.IsImage (PartialHomeomorph.symm e) t s ↔ PartialHomeomorph.IsImage e s t

theorem PartialHomeomorph.IsImage.mapsTo {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] {e : PartialHomeomorph α β} {s : Set α} {t : Set β} (h : PartialHomeomorph.IsImage e s t) : Set.MapsTo (↑e) (e.source ∩ s) (e.target ∩ t)

theorem PartialHomeomorph.IsImage.symm_mapsTo {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] {e : PartialHomeomorph α β} {s : Set α} {t : Set β} (h : PartialHomeomorph.IsImage e s t) : Set.MapsTo (↑(PartialHomeomorph.symm e)) (e.target ∩ t) (e.source ∩ s)

theorem PartialHomeomorph.IsImage.image_eq {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] {e : PartialHomeomorph α β} {s : Set α} {t : Set β} (h : PartialHomeomorph.IsImage e s t) : ↑e '' (e.source ∩ s) = e.target ∩ t

theorem PartialHomeomorph.IsImage.symm_image_eq {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] {e : PartialHomeomorph α β} {s : Set α} {t : Set β} (h : PartialHomeomorph.IsImage e s t) : ↑(PartialHomeomorph.symm e) '' (e.target ∩ t) = e.source ∩ s

theorem PartialHomeomorph.IsImage.iff_preimage_eq {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] {e : PartialHomeomorph α β} {s : Set α} {t : Set β} : PartialHomeomorph.IsImage e s t ↔ e.source ∩ ↑e ⁻¹' t = e.source ∩ s

theorem PartialHomeomorph.IsImage.preimage_eq {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] {e : PartialHomeomorph α β} {s : Set α} {t : Set β} : PartialHomeomorph.IsImage e s t → e.source ∩ ↑e ⁻¹' t = e.source ∩ s

Alias of the forward direction of PartialHomeomorph.IsImage.iff_preimage_eq.

theorem PartialHomeomorph.IsImage.of_preimage_eq {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] {e : PartialHomeomorph α β} {s : Set α} {t : Set β} : e.source ∩ ↑e ⁻¹' t = e.source ∩ s → PartialHomeomorph.IsImage e s t

Alias of the reverse direction of PartialHomeomorph.IsImage.iff_preimage_eq.

theorem PartialHomeomorph.IsImage.iff_symm_preimage_eq {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] {e : PartialHomeomorph α β} {s : Set α} {t : Set β} : PartialHomeomorph.IsImage e s t ↔ e.target ∩ ↑(PartialHomeomorph.symm e) ⁻¹' s = e.target ∩ t

theorem PartialHomeomorph.IsImage.of_symm_preimage_eq {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] {e : PartialHomeomorph α β} {s : Set α} {t : Set β} : e.target ∩ ↑(PartialHomeomorph.symm e) ⁻¹' s = e.target ∩ t → PartialHomeomorph.IsImage e s t

Alias of the reverse direction of PartialHomeomorph.IsImage.iff_symm_preimage_eq.

theorem PartialHomeomorph.IsImage.symm_preimage_eq {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] {e : PartialHomeomorph α β} {s : Set α} {t : Set β} : PartialHomeomorph.IsImage e s t → e.target ∩ ↑(PartialHomeomorph.symm e) ⁻¹' s = e.target ∩ t

Alias of the forward direction of PartialHomeomorph.IsImage.iff_symm_preimage_eq.

theorem PartialHomeomorph.IsImage.iff_symm_preimage_eq' {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] {e : PartialHomeomorph α β} {s : Set α} {t : Set β} : PartialHomeomorph.IsImage e s t ↔ e.target ∩ ↑(PartialHomeomorph.symm e) ⁻¹' (e.source ∩ s) = e.target ∩ t

theorem PartialHomeomorph.IsImage.symm_preimage_eq' {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] {e : PartialHomeomorph α β} {s : Set α} {t : Set β} : PartialHomeomorph.IsImage e s t → e.target ∩ ↑(PartialHomeomorph.symm e) ⁻¹' (e.source ∩ s) = e.target ∩ t

Alias of the forward direction of PartialHomeomorph.IsImage.iff_symm_preimage_eq'.

theorem PartialHomeomorph.IsImage.of_symm_preimage_eq' {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] {e : PartialHomeomorph α β} {s : Set α} {t : Set β} : e.target ∩ ↑(PartialHomeomorph.symm e) ⁻¹' (e.source ∩ s) = e.target ∩ t → PartialHomeomorph.IsImage e s t

Alias of the reverse direction of PartialHomeomorph.IsImage.iff_symm_preimage_eq'.

theorem PartialHomeomorph.IsImage.iff_preimage_eq' {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] {e : PartialHomeomorph α β} {s : Set α} {t : Set β} : PartialHomeomorph.IsImage e s t ↔ e.source ∩ ↑e ⁻¹' (e.target ∩ t) = e.source ∩ s

theorem PartialHomeomorph.IsImage.preimage_eq' {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] {e : PartialHomeomorph α β} {s : Set α} {t : Set β} : PartialHomeomorph.IsImage e s t → e.source ∩ ↑e ⁻¹' (e.target ∩ t) = e.source ∩ s

Alias of the forward direction of PartialHomeomorph.IsImage.iff_preimage_eq'.

theorem PartialHomeomorph.IsImage.of_preimage_eq' {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] {e : PartialHomeomorph α β} {s : Set α} {t : Set β} : e.source ∩ ↑e ⁻¹' (e.target ∩ t) = e.source ∩ s → PartialHomeomorph.IsImage e s t

Alias of the reverse direction of PartialHomeomorph.IsImage.iff_preimage_eq'.

theorem PartialHomeomorph.IsImage.of_image_eq {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] {e : PartialHomeomorph α β} {s : Set α} {t : Set β} (h : ↑e '' (e.source ∩ s) = e.target ∩ t) : PartialHomeomorph.IsImage e s t

theorem PartialHomeomorph.IsImage.of_symm_image_eq {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] {e : PartialHomeomorph α β} {s : Set α} {t : Set β} (h : ↑(PartialHomeomorph.symm e) '' (e.target ∩ t) = e.source ∩ s) : PartialHomeomorph.IsImage e s t

theorem PartialHomeomorph.IsImage.compl {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] {e : PartialHomeomorph α β} {s : Set α} {t : Set β} (h : PartialHomeomorph.IsImage e s t) : PartialHomeomorph.IsImage e sᶜ tᶜ

theorem PartialHomeomorph.IsImage.inter {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] {e : PartialHomeomorph α β} {s : Set α} {t : Set β} {s' : Set α} {t' : Set β} (h : PartialHomeomorph.IsImage e s t) (h' : PartialHomeomorph.IsImage e s' t') : PartialHomeomorph.IsImage e (s ∩ s') (t ∩ t')

theorem PartialHomeomorph.IsImage.union {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] {e : PartialHomeomorph α β} {s : Set α} {t : Set β} {s' : Set α} {t' : Set β} (h : PartialHomeomorph.IsImage e s t) (h' : PartialHomeomorph.IsImage e s' t') : PartialHomeomorph.IsImage e (s ∪ s') (t ∪ t')

theorem PartialHomeomorph.IsImage.diff {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] {e : PartialHomeomorph α β} {s : Set α} {t : Set β} {s' : Set α} {t' : Set β} (h : PartialHomeomorph.IsImage e s t) (h' : PartialHomeomorph.IsImage e s' t') : PartialHomeomorph.IsImage e (s \ s') (t \ t')

theorem PartialHomeomorph.IsImage.leftInvOn_piecewise {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] {e : PartialHomeomorph α β} {s : Set α} {t : Set β} {e' : PartialHomeomorph α β} [(i : α) → Decidable (i ∈ s)] [(i : β) → Decidable (i ∈ t)] (h : PartialHomeomorph.IsImage e s t) (h' : PartialHomeomorph.IsImage e' s t) : Set.LeftInvOn (Set.piecewise t ↑(PartialHomeomorph.symm e) ↑(PartialHomeomorph.symm e')) (Set.piecewise s ↑e ↑e') (Set.ite s e.source e'.source)

theorem PartialHomeomorph.IsImage.inter_eq_of_inter_eq_of_eqOn {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] {e : PartialHomeomorph α β} {s : Set α} {t : Set β} {e' : PartialHomeomorph α β} (h : PartialHomeomorph.IsImage e s t) (h' : PartialHomeomorph.IsImage e' s t) (hs : e.source ∩ s = e'.source ∩ s) (Heq : Set.EqOn (↑e) (↑e') (e.source ∩ s)) : e.target ∩ t = e'.target ∩ t

theorem PartialHomeomorph.IsImage.symm_eqOn_of_inter_eq_of_eqOn {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] {e : PartialHomeomorph α β} {s : Set α} {t : Set β} {e' : PartialHomeomorph α β} (h : PartialHomeomorph.IsImage e s t) (hs : e.source ∩ s = e'.source ∩ s) (Heq : Set.EqOn (↑e) (↑e') (e.source ∩ s)) : Set.EqOn (↑(PartialHomeomorph.symm e)) (↑(PartialHomeomorph.symm e')) (e.target ∩ t)

theorem PartialHomeomorph.IsImage.map_nhdsWithin_eq {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] {e : PartialHomeomorph α β} {s : Set α} {t : Set β} {x : α} (h : PartialHomeomorph.IsImage e s t) (hx : x ∈ e.source) : Filter.map (↑e) (nhdsWithin x s) = nhdsWithin (↑e x) t

theorem PartialHomeomorph.IsImage.closure {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] {e : PartialHomeomorph α β} {s : Set α} {t : Set β} (h : PartialHomeomorph.IsImage e s t) : PartialHomeomorph.IsImage e (closure s) (closure t)

theorem PartialHomeomorph.IsImage.interior {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] {e : PartialHomeomorph α β} {s : Set α} {t : Set β} (h : PartialHomeomorph.IsImage e s t) : PartialHomeomorph.IsImage e (interior s) (interior t)

theorem PartialHomeomorph.IsImage.frontier {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] {e : PartialHomeomorph α β} {s : Set α} {t : Set β} (h : PartialHomeomorph.IsImage e s t) : PartialHomeomorph.IsImage e (frontier s) (frontier t)

theorem PartialHomeomorph.IsImage.isOpen_iff {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] {e : PartialHomeomorph α β} {s : Set α} {t : Set β} (h : PartialHomeomorph.IsImage e s t) : IsOpen (e.source ∩ s) ↔ IsOpen (e.target ∩ t)

@[simp]

theorem PartialHomeomorph.IsImage.restr_toPartialEquiv {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] {e : PartialHomeomorph α β} {s : Set α} {t : Set β} (h : PartialHomeomorph.IsImage e s t) (hs : IsOpen (e.source ∩ s)) : (PartialHomeomorph.IsImage.restr h hs).toPartialEquiv = PartialEquiv.IsImage.restr (_ : PartialEquiv.IsImage e.toPartialEquiv s t)

def PartialHomeomorph.IsImage.restr {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] {e : PartialHomeomorph α β} {s : Set α} {t : Set β} (h : PartialHomeomorph.IsImage e s t) (hs : IsOpen (e.source ∩ s)) : PartialHomeomorph α β

Restrict a PartialHomeomorph to a pair of corresponding open sets.

Equations

• One or more equations did not get rendered due to their size.

theorem PartialHomeomorph.isImage_source_target {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] (e : PartialHomeomorph α β) : PartialHomeomorph.IsImage e e.source e.target

theorem PartialHomeomorph.isImage_source_target_of_disjoint {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] (e : PartialHomeomorph α β) (e' : PartialHomeomorph α β) (hs : Disjoint e.source e'.source) (ht : Disjoint e.target e'.target) : PartialHomeomorph.IsImage e e'.source e'.target

theorem PartialHomeomorph.preimage_interior {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] (e : PartialHomeomorph α β) (s : Set β) : e.source ∩ ↑e ⁻¹' interior s = e.source ∩ interior (↑e ⁻¹' s)

Preimage of interior or interior of preimage coincide for partial homeomorphisms, when restricted to the source.

theorem PartialHomeomorph.preimage_closure {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] (e : PartialHomeomorph α β) (s : Set β) : e.source ∩ ↑e ⁻¹' closure s = e.source ∩ closure (↑e ⁻¹' s)

theorem PartialHomeomorph.preimage_frontier {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] (e : PartialHomeomorph α β) (s : Set β) : e.source ∩ ↑e ⁻¹' frontier s = e.source ∩ frontier (↑e ⁻¹' s)

def PartialHomeomorph.ofContinuousOpenRestrict {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] (e : PartialEquiv α β) (hc : ContinuousOn (↑e) e.source) (ho : IsOpenMap (Set.restrict e.source ↑e)) (hs : IsOpen e.source) : PartialHomeomorph α β

A PartialEquiv with continuous open forward map and open source is a PartialHomeomorph.

Equations

• One or more equations did not get rendered due to their size.

def PartialHomeomorph.ofContinuousOpen {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] (e : PartialEquiv α β) (hc : ContinuousOn (↑e) e.source) (ho : IsOpenMap ↑e) (hs : IsOpen e.source) : PartialHomeomorph α β

A PartialEquiv with continuous open forward map and open source is a PartialHomeomorph.

Equations

• PartialHomeomorph.ofContinuousOpen e hc ho hs = PartialHomeomorph.ofContinuousOpenRestrict e hc (_ : IsOpenMap (Set.restrict e.source ↑e)) hs

def PartialHomeomorph.restrOpen {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] (e : PartialHomeomorph α β) (s : Set α) (hs : IsOpen s) : PartialHomeomorph α β

Restricting a partial homeomorphism e to e.source ∩ s when s is open. This is sometimes hard to use because of the openness assumption, but it has the advantage that when it can be used then its PartialEquiv is defeq to PartialEquiv.restr.

Equations

• PartialHomeomorph.restrOpen e s hs = PartialHomeomorph.IsImage.restr (_ : PartialHomeomorph.IsImage e s (↑(PartialHomeomorph.symm e) ⁻¹' s)) (_ : IsOpen (e.source ∩ s))

@[simp]

theorem PartialHomeomorph.restrOpen_toPartialEquiv {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] (e : PartialHomeomorph α β) (s : Set α) (hs : IsOpen s) : (PartialHomeomorph.restrOpen e s hs).toPartialEquiv = PartialEquiv.restr e.toPartialEquiv s

theorem PartialHomeomorph.restrOpen_source {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] (e : PartialHomeomorph α β) (s : Set α) (hs : IsOpen s) : (PartialHomeomorph.restrOpen e s hs).toPartialEquiv.source = e.source ∩ s

@[simp]

theorem PartialHomeomorph.restr_apply {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] (e : PartialHomeomorph α β) (s : Set α) : ↑(PartialHomeomorph.restr e s) = ↑e

@[simp]

theorem PartialHomeomorph.restr_symm_apply {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] (e : PartialHomeomorph α β) (s : Set α) : ↑(PartialHomeomorph.symm (PartialHomeomorph.restr e s)) = ↑(PartialHomeomorph.symm e)

theorem PartialHomeomorph.restr_source {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] (e : PartialHomeomorph α β) (s : Set α) : (PartialHomeomorph.restr e s).toPartialEquiv.source = e.source ∩ interior s

theorem PartialHomeomorph.restr_target {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] (e : PartialHomeomorph α β) (s : Set α) : (PartialHomeomorph.restr e s).toPartialEquiv.target = e.target ∩ ↑(PartialHomeomorph.symm e) ⁻¹' interior s

def PartialHomeomorph.restr {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] (e : PartialHomeomorph α β) (s : Set α) : PartialHomeomorph α β

Restricting a partial homeomorphism e to e.source ∩ interior s. We use the interior to make sure that the restriction is well defined whatever the set s, since partial homeomorphisms are by definition defined on open sets. In applications where s is open, this coincides with the restriction of partial equivalences

Equations

• PartialHomeomorph.restr e s = PartialHomeomorph.restrOpen e (interior s) (_ : IsOpen (interior s))

@[simp]

theorem PartialHomeomorph.restr_toPartialEquiv {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] (e : PartialHomeomorph α β) (s : Set α) : (PartialHomeomorph.restr e s).toPartialEquiv = PartialEquiv.restr e.toPartialEquiv (interior s)

theorem PartialHomeomorph.restr_source' {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] (e : PartialHomeomorph α β) (s : Set α) (hs : IsOpen s) : (PartialHomeomorph.restr e s).toPartialEquiv.source = e.source ∩ s

theorem PartialHomeomorph.restr_toPartialEquiv' {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] (e : PartialHomeomorph α β) (s : Set α) (hs : IsOpen s) : (PartialHomeomorph.restr e s).toPartialEquiv = PartialEquiv.restr e.toPartialEquiv s

theorem PartialHomeomorph.restr_eq_of_source_subset {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] {e : PartialHomeomorph α β} {s : Set α} (h : e.source ⊆ s) : PartialHomeomorph.restr e s = e

@[simp]

theorem PartialHomeomorph.restr_univ {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] {e : PartialHomeomorph α β} : PartialHomeomorph.restr e Set.univ = e

theorem PartialHomeomorph.restr_source_inter {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] (e : PartialHomeomorph α β) (s : Set α) : PartialHomeomorph.restr e (e.source ∩ s) = PartialHomeomorph.restr e s

@[simp]

theorem PartialHomeomorph.refl_apply (α : Type u_5) [TopologicalSpace α] : ↑(PartialHomeomorph.refl α) = id

theorem PartialHomeomorph.refl_target (α : Type u_5) [TopologicalSpace α] : (PartialHomeomorph.refl α).toPartialEquiv.target = Set.univ

theorem PartialHomeomorph.refl_source (α : Type u_5) [TopologicalSpace α] : (PartialHomeomorph.refl α).toPartialEquiv.source = Set.univ

def PartialHomeomorph.refl (α : Type u_5) [TopologicalSpace α] : PartialHomeomorph α α

The identity on the whole space as a partial homeomorphism.

Equations

• PartialHomeomorph.refl α = Homeomorph.toPartialHomeomorph (Homeomorph.refl α)

@[simp]

theorem PartialHomeomorph.refl_localEquiv {α : Type u_1} [TopologicalSpace α] : (PartialHomeomorph.refl α).toPartialEquiv = PartialEquiv.refl α

@[simp]

theorem PartialHomeomorph.refl_symm {α : Type u_1} [TopologicalSpace α] : PartialHomeomorph.symm (PartialHomeomorph.refl α) = PartialHomeomorph.refl α

@[simp]

theorem PartialHomeomorph.ofSet_apply {α : Type u_1} [TopologicalSpace α] (s : Set α) (hs : IsOpen s) : ↑(PartialHomeomorph.ofSet s hs) = id

theorem PartialHomeomorph.ofSet_source {α : Type u_1} [TopologicalSpace α] (s : Set α) (hs : IsOpen s) : (PartialHomeomorph.ofSet s hs).toPartialEquiv.source = s

theorem PartialHomeomorph.ofSet_target {α : Type u_1} [TopologicalSpace α] (s : Set α) (hs : IsOpen s) : (PartialHomeomorph.ofSet s hs).toPartialEquiv.target = s

def PartialHomeomorph.ofSet {α : Type u_1} [TopologicalSpace α] (s : Set α) (hs : IsOpen s) : PartialHomeomorph α α

The identity partial equivalence on a set s

Equations

• One or more equations did not get rendered due to their size.

@[simp]

theorem PartialHomeomorph.ofSet_toPartialEquiv {α : Type u_1} [TopologicalSpace α] {s : Set α} (hs : IsOpen s) : (PartialHomeomorph.ofSet s hs).toPartialEquiv = PartialEquiv.ofSet s

@[simp]

theorem PartialHomeomorph.ofSet_symm {α : Type u_1} [TopologicalSpace α] {s : Set α} (hs : IsOpen s) : PartialHomeomorph.symm (PartialHomeomorph.ofSet s hs) = PartialHomeomorph.ofSet s hs

@[simp]

theorem PartialHomeomorph.ofSet_univ_eq_refl {α : Type u_1} [TopologicalSpace α] : PartialHomeomorph.ofSet Set.univ (_ : IsOpen Set.univ) = PartialHomeomorph.refl α

@[simp]

theorem PartialHomeomorph.trans'_apply {α : Type u_1} {β : Type u_2} {γ : Type u_3} [TopologicalSpace α] [TopologicalSpace β] [TopologicalSpace γ] (e : PartialHomeomorph α β) (e' : PartialHomeomorph β γ) (h : e.target = e'.source) : ∀ (a : α), ↑(PartialHomeomorph.trans' e e' h) a = ↑e' (↑e a)

@[simp]

theorem PartialHomeomorph.trans'_symm_apply {α : Type u_1} {β : Type u_2} {γ : Type u_3} [TopologicalSpace α] [TopologicalSpace β] [TopologicalSpace γ] (e : PartialHomeomorph α β) (e' : PartialHomeomorph β γ) (h : e.target = e'.source) : ∀ (a : γ), ↑(PartialHomeomorph.symm (PartialHomeomorph.trans' e e' h)) a = ↑(PartialHomeomorph.symm e) (↑(PartialHomeomorph.symm e') a)

@[simp]

theorem PartialHomeomorph.trans'_toPartialEquiv {α : Type u_1} {β : Type u_2} {γ : Type u_3} [TopologicalSpace α] [TopologicalSpace β] [TopologicalSpace γ] (e : PartialHomeomorph α β) (e' : PartialHomeomorph β γ) (h : e.target = e'.source) : (PartialHomeomorph.trans' e e' h).toPartialEquiv = PartialEquiv.trans' e.toPartialEquiv e'.toPartialEquiv h

theorem PartialHomeomorph.trans'_source {α : Type u_1} {β : Type u_2} {γ : Type u_3} [TopologicalSpace α] [TopologicalSpace β] [TopologicalSpace γ] (e : PartialHomeomorph α β) (e' : PartialHomeomorph β γ) (h : e.target = e'.source) : (PartialHomeomorph.trans' e e' h).toPartialEquiv.source = e.source

theorem PartialHomeomorph.trans'_target {α : Type u_1} {β : Type u_2} {γ : Type u_3} [TopologicalSpace α] [TopologicalSpace β] [TopologicalSpace γ] (e : PartialHomeomorph α β) (e' : PartialHomeomorph β γ) (h : e.target = e'.source) : (PartialHomeomorph.trans' e e' h).toPartialEquiv.target = e'.target

def PartialHomeomorph.trans' {α : Type u_1} {β : Type u_2} {γ : Type u_3} [TopologicalSpace α] [TopologicalSpace β] [TopologicalSpace γ] (e : PartialHomeomorph α β) (e' : PartialHomeomorph β γ) (h : e.target = e'.source) : PartialHomeomorph α γ

Composition of two partial homeomorphisms when the target of the first and the source of the second coincide.

Equations

• One or more equations did not get rendered due to their size.

def PartialHomeomorph.trans {α : Type u_1} {β : Type u_2} {γ : Type u_3} [TopologicalSpace α] [TopologicalSpace β] [TopologicalSpace γ] (e : PartialHomeomorph α β) (e' : PartialHomeomorph β γ) : PartialHomeomorph α γ

Composing two partial homeomorphisms, by restricting to the maximal domain where their composition is well defined.

Equations

• One or more equations did not get rendered due to their size.

@[simp]

theorem PartialHomeomorph.trans_toPartialEquiv {α : Type u_1} {β : Type u_2} {γ : Type u_3} [TopologicalSpace α] [TopologicalSpace β] [TopologicalSpace γ] (e : PartialHomeomorph α β) (e' : PartialHomeomorph β γ) : (PartialHomeomorph.trans e e').toPartialEquiv = PartialEquiv.trans e.toPartialEquiv e'.toPartialEquiv

@[simp]

theorem PartialHomeomorph.coe_trans {α : Type u_1} {β : Type u_2} {γ : Type u_3} [TopologicalSpace α] [TopologicalSpace β] [TopologicalSpace γ] (e : PartialHomeomorph α β) (e' : PartialHomeomorph β γ) : ↑(PartialHomeomorph.trans e e') = ↑e' ∘ ↑e

@[simp]

theorem PartialHomeomorph.coe_trans_symm {α : Type u_1} {β : Type u_2} {γ : Type u_3} [TopologicalSpace α] [TopologicalSpace β] [TopologicalSpace γ] (e : PartialHomeomorph α β) (e' : PartialHomeomorph β γ) : ↑(PartialHomeomorph.symm (PartialHomeomorph.trans e e')) = ↑(PartialHomeomorph.symm e) ∘ ↑(PartialHomeomorph.symm e')

theorem PartialHomeomorph.trans_apply {α : Type u_1} {β : Type u_2} {γ : Type u_3} [TopologicalSpace α] [TopologicalSpace β] [TopologicalSpace γ] (e : PartialHomeomorph α β) (e' : PartialHomeomorph β γ) {x : α} : ↑(PartialHomeomorph.trans e e') x = ↑e' (↑e x)

theorem PartialHomeomorph.trans_symm_eq_symm_trans_symm {α : Type u_1} {β : Type u_2} {γ : Type u_3} [TopologicalSpace α] [TopologicalSpace β] [TopologicalSpace γ] (e : PartialHomeomorph α β) (e' : PartialHomeomorph β γ) : PartialHomeomorph.symm (PartialHomeomorph.trans e e') = PartialHomeomorph.trans (PartialHomeomorph.symm e') (PartialHomeomorph.symm e)

theorem PartialHomeomorph.trans_source {α : Type u_1} {β : Type u_2} {γ : Type u_3} [TopologicalSpace α] [TopologicalSpace β] [TopologicalSpace γ] (e : PartialHomeomorph α β) (e' : PartialHomeomorph β γ) : (PartialHomeomorph.trans e e').toPartialEquiv.source = e.source ∩ ↑e ⁻¹' e'.source

theorem PartialHomeomorph.trans_source' {α : Type u_1} {β : Type u_2} {γ : Type u_3} [TopologicalSpace α] [TopologicalSpace β] [TopologicalSpace γ] (e : PartialHomeomorph α β) (e' : PartialHomeomorph β γ) : (PartialHomeomorph.trans e e').toPartialEquiv.source = e.source ∩ ↑e ⁻¹' (e.target ∩ e'.source)

theorem PartialHomeomorph.trans_source'' {α : Type u_1} {β : Type u_2} {γ : Type u_3} [TopologicalSpace α] [TopologicalSpace β] [TopologicalSpace γ] (e : PartialHomeomorph α β) (e' : PartialHomeomorph β γ) : (PartialHomeomorph.trans e e').toPartialEquiv.source = ↑(PartialHomeomorph.symm e) '' (e.target ∩ e'.source)

theorem PartialHomeomorph.image_trans_source {α : Type u_1} {β : Type u_2} {γ : Type u_3} [TopologicalSpace α] [TopologicalSpace β] [TopologicalSpace γ] (e : PartialHomeomorph α β) (e' : PartialHomeomorph β γ) : ↑e '' (PartialHomeomorph.trans e e').toPartialEquiv.source = e.target ∩ e'.source

theorem PartialHomeomorph.trans_target {α : Type u_1} {β : Type u_2} {γ : Type u_3} [TopologicalSpace α] [TopologicalSpace β] [TopologicalSpace γ] (e : PartialHomeomorph α β) (e' : PartialHomeomorph β γ) : (PartialHomeomorph.trans e e').toPartialEquiv.target = e'.target ∩ ↑(PartialHomeomorph.symm e') ⁻¹' e.target

theorem PartialHomeomorph.trans_target' {α : Type u_1} {β : Type u_2} {γ : Type u_3} [TopologicalSpace α] [TopologicalSpace β] [TopologicalSpace γ] (e : PartialHomeomorph α β) (e' : PartialHomeomorph β γ) : (PartialHomeomorph.trans e e').toPartialEquiv.target = e'.target ∩ ↑(PartialHomeomorph.symm e') ⁻¹' (e'.source ∩ e.target)

theorem PartialHomeomorph.trans_target'' {α : Type u_1} {β : Type u_2} {γ : Type u_3} [TopologicalSpace α] [TopologicalSpace β] [TopologicalSpace γ] (e : PartialHomeomorph α β) (e' : PartialHomeomorph β γ) : (PartialHomeomorph.trans e e').toPartialEquiv.target = ↑e' '' (e'.source ∩ e.target)

theorem PartialHomeomorph.inv_image_trans_target {α : Type u_1} {β : Type u_2} {γ : Type u_3} [TopologicalSpace α] [TopologicalSpace β] [TopologicalSpace γ] (e : PartialHomeomorph α β) (e' : PartialHomeomorph β γ) : ↑(PartialHomeomorph.symm e') '' (PartialHomeomorph.trans e e').toPartialEquiv.target = e'.source ∩ e.target

theorem PartialHomeomorph.trans_assoc {α : Type u_1} {β : Type u_2} {γ : Type u_3} {δ : Type u_4} [TopologicalSpace α] [TopologicalSpace β] [TopologicalSpace γ] [TopologicalSpace δ] (e : PartialHomeomorph α β) (e' : PartialHomeomorph β γ) (e'' : PartialHomeomorph γ δ) : PartialHomeomorph.trans (PartialHomeomorph.trans e e') e'' = PartialHomeomorph.trans e (PartialHomeomorph.trans e' e'')

@[simp]

theorem PartialHomeomorph.trans_refl {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] (e : PartialHomeomorph α β) : PartialHomeomorph.trans e (PartialHomeomorph.refl β) = e

@[simp]

theorem PartialHomeomorph.refl_trans {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] (e : PartialHomeomorph α β) : PartialHomeomorph.trans (PartialHomeomorph.refl α) e = e

theorem PartialHomeomorph.trans_ofSet {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] (e : PartialHomeomorph α β) {s : Set β} (hs : IsOpen s) : PartialHomeomorph.trans e (PartialHomeomorph.ofSet s hs) = PartialHomeomorph.restr e (↑e ⁻¹' s)

theorem PartialHomeomorph.trans_of_set' {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] (e : PartialHomeomorph α β) {s : Set β} (hs : IsOpen s) : PartialHomeomorph.trans e (PartialHomeomorph.ofSet s hs) = PartialHomeomorph.restr e (e.source ∩ ↑e ⁻¹' s)

theorem PartialHomeomorph.ofSet_trans {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] (e : PartialHomeomorph α β) {s : Set α} (hs : IsOpen s) : PartialHomeomorph.trans (PartialHomeomorph.ofSet s hs) e = PartialHomeomorph.restr e s

theorem PartialHomeomorph.ofSet_trans' {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] (e : PartialHomeomorph α β) {s : Set α} (hs : IsOpen s) : PartialHomeomorph.trans (PartialHomeomorph.ofSet s hs) e = PartialHomeomorph.restr e (e.source ∩ s)

@[simp]

theorem PartialHomeomorph.ofSet_trans_ofSet {α : Type u_1} [TopologicalSpace α] {s : Set α} (hs : IsOpen s) {s' : Set α} (hs' : IsOpen s') : PartialHomeomorph.trans (PartialHomeomorph.ofSet s hs) (PartialHomeomorph.ofSet s' hs') = PartialHomeomorph.ofSet (s ∩ s') (_ : IsOpen (s ∩ s'))

theorem PartialHomeomorph.restr_trans {α : Type u_1} {β : Type u_2} {γ : Type u_3} [TopologicalSpace α] [TopologicalSpace β] [TopologicalSpace γ] (e : PartialHomeomorph α β) (e' : PartialHomeomorph β γ) (s : Set α) : PartialHomeomorph.trans (PartialHomeomorph.restr e s) e' = PartialHomeomorph.restr (PartialHomeomorph.trans e e') s

def PartialHomeomorph.EqOnSource {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] (e : PartialHomeomorph α β) (e' : PartialHomeomorph α β) : Prop

EqOnSource e e' means that e and e' have the same source, and coincide there. They should really be considered the same partial equivalence.

Equations

• PartialHomeomorph.EqOnSource e e' = (e.source = e'.source ∧ Set.EqOn (↑e) (↑e') e.source)

theorem PartialHomeomorph.eqOnSource_iff {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] (e : PartialHomeomorph α β) (e' : PartialHomeomorph α β) : PartialHomeomorph.EqOnSource e e' ↔ PartialEquiv.EqOnSource e.toPartialEquiv e'.toPartialEquiv

instance PartialHomeomorph.eqOnSourceSetoid {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] : Setoid (PartialHomeomorph α β)

EqOnSource is an equivalence relation.

Equations

• One or more equations did not get rendered due to their size.

theorem PartialHomeomorph.eqOnSource_refl {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] (e : PartialHomeomorph α β) : e ≈ e

theorem PartialHomeomorph.EqOnSource.symm' {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] {e : PartialHomeomorph α β} {e' : PartialHomeomorph α β} (h : e ≈ e') : PartialHomeomorph.symm e ≈ PartialHomeomorph.symm e'

If two partial homeomorphisms are equivalent, so are their inverses.

theorem PartialHomeomorph.EqOnSource.source_eq {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] {e : PartialHomeomorph α β} {e' : PartialHomeomorph α β} (h : e ≈ e') : e.source = e'.source

Two equivalent partial homeomorphisms have the same source.

theorem PartialHomeomorph.EqOnSource.target_eq {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] {e : PartialHomeomorph α β} {e' : PartialHomeomorph α β} (h : e ≈ e') : e.target = e'.target

Two equivalent partial homeomorphisms have the same target.

theorem PartialHomeomorph.EqOnSource.eqOn {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] {e : PartialHomeomorph α β} {e' : PartialHomeomorph α β} (h : e ≈ e') : Set.EqOn (↑e) (↑e') e.source

Two equivalent partial homeomorphisms have coinciding toFun on the source

theorem PartialHomeomorph.EqOnSource.symm_eqOn_target {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] {e : PartialHomeomorph α β} {e' : PartialHomeomorph α β} (h : e ≈ e') : Set.EqOn (↑(PartialHomeomorph.symm e)) (↑(PartialHomeomorph.symm e')) e.target

Two equivalent partial homeomorphisms have coinciding invFun on the target

theorem PartialHomeomorph.EqOnSource.trans' {α : Type u_1} {β : Type u_2} {γ : Type u_3} [TopologicalSpace α] [TopologicalSpace β] [TopologicalSpace γ] {e : PartialHomeomorph α β} {e' : PartialHomeomorph α β} {f : PartialHomeomorph β γ} {f' : PartialHomeomorph β γ} (he : e ≈ e') (hf : f ≈ f') : PartialHomeomorph.trans e f ≈ PartialHomeomorph.trans e' f'

Composition of partial homeomorphisms respects equivalence.

theorem PartialHomeomorph.EqOnSource.restr {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] {e : PartialHomeomorph α β} {e' : PartialHomeomorph α β} (he : e ≈ e') (s : Set α) : PartialHomeomorph.restr e s ≈ PartialHomeomorph.restr e' s

Restriction of partial homeomorphisms respects equivalence

theorem PartialHomeomorph.Set.EqOn.restr_eqOn_source {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] {e : PartialHomeomorph α β} {e' : PartialHomeomorph α β} (h : Set.EqOn (↑e) (↑e') (e.source ∩ e'.source)) : PartialHomeomorph.restr e e'.source ≈ PartialHomeomorph.restr e' e.source

theorem PartialHomeomorph.self_trans_symm {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] (e : PartialHomeomorph α β) : PartialHomeomorph.trans e (PartialHomeomorph.symm e) ≈ PartialHomeomorph.ofSet e.source (_ : IsOpen e.source)

Composition of a partial homeomorphism and its inverse is equivalent to the restriction of the identity to the source

theorem PartialHomeomorph.symm_trans_self {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] (e : PartialHomeomorph α β) : PartialHomeomorph.trans (PartialHomeomorph.symm e) e ≈ PartialHomeomorph.ofSet e.target (_ : IsOpen e.target)

theorem PartialHomeomorph.eq_of_eqOnSource_univ {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] {e : PartialHomeomorph α β} {e' : PartialHomeomorph α β} (h : e ≈ e') (s : e.source = Set.univ) (t : e.target = Set.univ) : e = e'

@[simp]

theorem PartialHomeomorph.prod_apply {α : Type u_1} {β : Type u_2} {γ : Type u_3} {δ : Type u_4} [TopologicalSpace α] [TopologicalSpace β] [TopologicalSpace γ] [TopologicalSpace δ] (e : PartialHomeomorph α β) (e' : PartialHomeomorph γ δ) : ↑(PartialHomeomorph.prod e e') = fun (p : α × γ) => (↑e p.1, ↑e' p.2)

@[simp]

theorem PartialHomeomorph.prod_toPartialEquiv {α : Type u_1} {β : Type u_2} {γ : Type u_3} {δ : Type u_4} [TopologicalSpace α] [TopologicalSpace β] [TopologicalSpace γ] [TopologicalSpace δ] (e : PartialHomeomorph α β) (e' : PartialHomeomorph γ δ) : (PartialHomeomorph.prod e e').toPartialEquiv = PartialEquiv.prod e.toPartialEquiv e'.toPartialEquiv

theorem PartialHomeomorph.prod_source {α : Type u_1} {β : Type u_2} {γ : Type u_3} {δ : Type u_4} [TopologicalSpace α] [TopologicalSpace β] [TopologicalSpace γ] [TopologicalSpace δ] (e : PartialHomeomorph α β) (e' : PartialHomeomorph γ δ) : (PartialHomeomorph.prod e e').toPartialEquiv.source = e.source ×ˢ e'.source

theorem PartialHomeomorph.prod_target {α : Type u_1} {β : Type u_2} {γ : Type u_3} {δ : Type u_4} [TopologicalSpace α] [TopologicalSpace β] [TopologicalSpace γ] [TopologicalSpace δ] (e : PartialHomeomorph α β) (e' : PartialHomeomorph γ δ) : (PartialHomeomorph.prod e e').toPartialEquiv.target = e.target ×ˢ e'.target

theorem PartialHomeomorph.prod_symm_apply {α : Type u_1} {β : Type u_2} {γ : Type u_3} {δ : Type u_4} [TopologicalSpace α] [TopologicalSpace β] [TopologicalSpace γ] [TopologicalSpace δ] (e : PartialHomeomorph α β) (e' : PartialHomeomorph γ δ) (p : β × δ) : ↑(PartialHomeomorph.symm (PartialHomeomorph.prod e e')) p = (↑(PartialHomeomorph.symm e) p.1, ↑(PartialHomeomorph.symm e') p.2)

def PartialHomeomorph.prod {α : Type u_1} {β : Type u_2} {γ : Type u_3} {δ : Type u_4} [TopologicalSpace α] [TopologicalSpace β] [TopologicalSpace γ] [TopologicalSpace δ] (e : PartialHomeomorph α β) (e' : PartialHomeomorph γ δ) : PartialHomeomorph (α × γ) (β × δ)

The product of two partial homeomorphisms, as a partial homeomorphism on the product space.

Equations

• One or more equations did not get rendered due to their size.

@[simp]

theorem PartialHomeomorph.prod_symm {α : Type u_1} {β : Type u_2} {γ : Type u_3} {δ : Type u_4} [TopologicalSpace α] [TopologicalSpace β] [TopologicalSpace γ] [TopologicalSpace δ] (e : PartialHomeomorph α β) (e' : PartialHomeomorph γ δ) : PartialHomeomorph.symm (PartialHomeomorph.prod e e') = PartialHomeomorph.prod (PartialHomeomorph.symm e) (PartialHomeomorph.symm e')

@[simp]

theorem PartialHomeomorph.refl_prod_refl {α : Type u_5} {β : Type u_6} [TopologicalSpace α] [TopologicalSpace β] : PartialHomeomorph.prod (PartialHomeomorph.refl α) (PartialHomeomorph.refl β) = PartialHomeomorph.refl (α × β)

@[simp]

theorem PartialHomeomorph.prod_trans {α : Type u_1} {β : Type u_2} {γ : Type u_3} {δ : Type u_4} [TopologicalSpace α] [TopologicalSpace β] [TopologicalSpace γ] [TopologicalSpace δ] {η : Type u_5} {ε : Type u_6} [TopologicalSpace η] [TopologicalSpace ε] (e : PartialHomeomorph α β) (f : PartialHomeomorph β γ) (e' : PartialHomeomorph δ η) (f' : PartialHomeomorph η ε) : PartialHomeomorph.trans (PartialHomeomorph.prod e e') (PartialHomeomorph.prod f f') = PartialHomeomorph.prod (PartialHomeomorph.trans e f) (PartialHomeomorph.trans e' f')

theorem PartialHomeomorph.prod_eq_prod_of_nonempty {α : Type u_1} {β : Type u_2} {γ : Type u_3} {δ : Type u_4} [TopologicalSpace α] [TopologicalSpace β] [TopologicalSpace γ] [TopologicalSpace δ] {e₁ : PartialHomeomorph α β} {e₁' : PartialHomeomorph α β} {e₂ : PartialHomeomorph γ δ} {e₂' : PartialHomeomorph γ δ} (h : Set.Nonempty (PartialHomeomorph.prod e₁ e₂).toPartialEquiv.source) : PartialHomeomorph.prod e₁ e₂ = PartialHomeomorph.prod e₁' e₂' ↔ e₁ = e₁' ∧ e₂ = e₂'

theorem PartialHomeomorph.prod_eq_prod_of_nonempty' {α : Type u_1} {β : Type u_2} {γ : Type u_3} {δ : Type u_4} [TopologicalSpace α] [TopologicalSpace β] [TopologicalSpace γ] [TopologicalSpace δ] {e₁ : PartialHomeomorph α β} {e₁' : PartialHomeomorph α β} {e₂ : PartialHomeomorph γ δ} {e₂' : PartialHomeomorph γ δ} (h : Set.Nonempty (PartialHomeomorph.prod e₁' e₂').toPartialEquiv.source) : PartialHomeomorph.prod e₁ e₂ = PartialHomeomorph.prod e₁' e₂' ↔ e₁ = e₁' ∧ e₂ = e₂'

@[simp]

theorem PartialHomeomorph.piecewise_toPartialEquiv {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] (e : PartialHomeomorph α β) (e' : PartialHomeomorph α β) (s : Set α) (t : Set β) [(x : α) → Decidable (x ∈ s)] [(y : β) → Decidable (y ∈ t)] (H : PartialHomeomorph.IsImage e s t) (H' : PartialHomeomorph.IsImage e' s t) (Hs : e.source ∩ frontier s = e'.source ∩ frontier s) (Heq : Set.EqOn (↑e) (↑e') (e.source ∩ frontier s)) : (PartialHomeomorph.piecewise e e' s t H H' Hs Heq).toPartialEquiv = PartialEquiv.piecewise e.toPartialEquiv e'.toPartialEquiv s t H H'

@[simp]

theorem PartialHomeomorph.piecewise_apply {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] (e : PartialHomeomorph α β) (e' : PartialHomeomorph α β) (s : Set α) (t : Set β) [(x : α) → Decidable (x ∈ s)] [(y : β) → Decidable (y ∈ t)] (H : PartialHomeomorph.IsImage e s t) (H' : PartialHomeomorph.IsImage e' s t) (Hs : e.source ∩ frontier s = e'.source ∩ frontier s) (Heq : Set.EqOn (↑e) (↑e') (e.source ∩ frontier s)) : ↑(PartialHomeomorph.piecewise e e' s t H H' Hs Heq) = Set.piecewise s ↑e ↑e'

def PartialHomeomorph.piecewise {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] (e : PartialHomeomorph α β) (e' : PartialHomeomorph α β) (s : Set α) (t : Set β) [(x : α) → Decidable (x ∈ s)] [(y : β) → Decidable (y ∈ t)] (H : PartialHomeomorph.IsImage e s t) (H' : PartialHomeomorph.IsImage e' s t) (Hs : e.source ∩ frontier s = e'.source ∩ frontier s) (Heq : Set.EqOn (↑e) (↑e') (e.source ∩ frontier s)) : PartialHomeomorph α β

Combine two PartialHomeomorphs using Set.piecewise. The source of the new PartialHomeomorph is s.ite e.source e'.source = e.source ∩ s ∪ e'.source \ s, and similarly for target. The function sends e.source ∩ s to e.target ∩ t using e and e'.source \ s to e'.target \ t using e', and similarly for the inverse function. To ensure the maps toFun and invFun are inverse of each other on the new source and target, the definition assumes that the sets s and t are related both by e.is_image and e'.is_image. To ensure that the new maps are continuous on source/target, it also assumes that e.source and e'.source meet frontier s on the same set and e x = e' x on this intersection.

Equations

• One or more equations did not get rendered due to their size.

@[simp]

theorem PartialHomeomorph.symm_piecewise {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] (e : PartialHomeomorph α β) (e' : PartialHomeomorph α β) {s : Set α} {t : Set β} [(x : α) → Decidable (x ∈ s)] [(y : β) → Decidable (y ∈ t)] (H : PartialHomeomorph.IsImage e s t) (H' : PartialHomeomorph.IsImage e' s t) (Hs : e.source ∩ frontier s = e'.source ∩ frontier s) (Heq : Set.EqOn (↑e) (↑e') (e.source ∩ frontier s)) : PartialHomeomorph.symm (PartialHomeomorph.piecewise e e' s t H H' Hs Heq) = PartialHomeomorph.piecewise (PartialHomeomorph.symm e) (PartialHomeomorph.symm e') t s (_ : PartialHomeomorph.IsImage (PartialHomeomorph.symm e) t s) (_ : PartialHomeomorph.IsImage (PartialHomeomorph.symm e') t s) (_ : e.target ∩ frontier t = e'.target ∩ frontier t) (_ : Set.EqOn (↑(PartialHomeomorph.symm e)) (↑(PartialHomeomorph.symm e')) (e.target ∩ frontier t))

def PartialHomeomorph.disjointUnion {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] (e : PartialHomeomorph α β) (e' : PartialHomeomorph α β) [(x : α) → Decidable (x ∈ e.source)] [(y : β) → Decidable (y ∈ e.target)] (Hs : Disjoint e.source e'.source) (Ht : Disjoint e.target e'.target) : PartialHomeomorph α β

Combine two PartialHomeomorphs with disjoint sources and disjoint targets. We reuse PartialHomeomorph.piecewise then override toPartialEquiv to PartialEquiv.disjointUnion. This way we have better definitional equalities for source and target.

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@[simp]

theorem PartialHomeomorph.pi_toPartialEquiv {ι : Type u_5} [Fintype ι] {Xi : ι → Type u_6} {Yi : ι → Type u_7} [(i : ι) → TopologicalSpace (Xi i)] [(i : ι) → TopologicalSpace (Yi i)] (ei : (i : ι) → PartialHomeomorph (Xi i) (Yi i)) : (PartialHomeomorph.pi ei).toPartialEquiv = PartialEquiv.pi fun (i : ι) => (ei i).toPartialEquiv

def PartialHomeomorph.pi {ι : Type u_5} [Fintype ι] {Xi : ι → Type u_6} {Yi : ι → Type u_7} [(i : ι) → TopologicalSpace (Xi i)] [(i : ι) → TopologicalSpace (Yi i)] (ei : (i : ι) → PartialHomeomorph (Xi i) (Yi i)) : PartialHomeomorph ((i : ι) → Xi i) ((i : ι) → Yi i)

The product of a finite family of PartialHomeomorphs.

Equations

• One or more equations did not get rendered due to their size.

theorem PartialHomeomorph.continuousWithinAt_iff_continuousWithinAt_comp_right {α : Type u_1} {β : Type u_2} {γ : Type u_3} [TopologicalSpace α] [TopologicalSpace β] [TopologicalSpace γ] (e : PartialHomeomorph α β) {f : β → γ} {s : Set β} {x : β} (h : x ∈ e.target) : ContinuousWithinAt f s x ↔ ContinuousWithinAt (f ∘ ↑e) (↑e ⁻¹' s) (↑(PartialHomeomorph.symm e) x)

Continuity within a set at a point can be read under right composition with a local homeomorphism, if the point is in its target

theorem PartialHomeomorph.continuousAt_iff_continuousAt_comp_right {α : Type u_1} {β : Type u_2} {γ : Type u_3} [TopologicalSpace α] [TopologicalSpace β] [TopologicalSpace γ] (e : PartialHomeomorph α β) {f : β → γ} {x : β} (h : x ∈ e.target) : ContinuousAt f x ↔ ContinuousAt (f ∘ ↑e) (↑(PartialHomeomorph.symm e) x)

Continuity at a point can be read under right composition with a partial homeomorphism, if the point is in its target

theorem PartialHomeomorph.continuousOn_iff_continuousOn_comp_right {α : Type u_1} {β : Type u_2} {γ : Type u_3} [TopologicalSpace α] [TopologicalSpace β] [TopologicalSpace γ] (e : PartialHomeomorph α β) {f : β → γ} {s : Set β} (h : s ⊆ e.target) : ContinuousOn f s ↔ ContinuousOn (f ∘ ↑e) (e.source ∩ ↑e ⁻¹' s)

A function is continuous on a set if and only if its composition with a partial homeomorphism on the right is continuous on the corresponding set.

theorem PartialHomeomorph.continuousWithinAt_iff_continuousWithinAt_comp_left {α : Type u_1} {β : Type u_2} {γ : Type u_3} [TopologicalSpace α] [TopologicalSpace β] [TopologicalSpace γ] (e : PartialHomeomorph α β) {f : γ → α} {s : Set γ} {x : γ} (hx : f x ∈ e.source) (h : f ⁻¹' e.source ∈ nhdsWithin x s) : ContinuousWithinAt f s x ↔ ContinuousWithinAt (↑e ∘ f) s x

Continuity within a set at a point can be read under left composition with a local homeomorphism if a neighborhood of the initial point is sent to the source of the local homeomorphism

theorem PartialHomeomorph.continuousAt_iff_continuousAt_comp_left {α : Type u_1} {β : Type u_2} {γ : Type u_3} [TopologicalSpace α] [TopologicalSpace β] [TopologicalSpace γ] (e : PartialHomeomorph α β) {f : γ → α} {x : γ} (h : f ⁻¹' e.source ∈ nhds x) : ContinuousAt f x ↔ ContinuousAt (↑e ∘ f) x

Continuity at a point can be read under left composition with a partial homeomorphism if a neighborhood of the initial point is sent to the source of the partial homeomorphism

theorem PartialHomeomorph.continuousOn_iff_continuousOn_comp_left {α : Type u_1} {β : Type u_2} {γ : Type u_3} [TopologicalSpace α] [TopologicalSpace β] [TopologicalSpace γ] (e : PartialHomeomorph α β) {f : γ → α} {s : Set γ} (h : s ⊆ f ⁻¹' e.source) : ContinuousOn f s ↔ ContinuousOn (↑e ∘ f) s

A function is continuous on a set if and only if its composition with a partial homeomorphism on the left is continuous on the corresponding set.

theorem PartialHomeomorph.continuous_iff_continuous_comp_left {α : Type u_1} {β : Type u_2} {γ : Type u_3} [TopologicalSpace α] [TopologicalSpace β] [TopologicalSpace γ] (e : PartialHomeomorph α β) {f : γ → α} (h : f ⁻¹' e.source = Set.univ) : Continuous f ↔ Continuous (↑e ∘ f)

A function is continuous if and only if its composition with a partial homeomorphism on the left is continuous and its image is contained in the source.

@[simp]

theorem PartialHomeomorph.homeomorphOfImageSubsetSource_apply {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] (e : PartialHomeomorph α β) {s : Set α} {t : Set β} (hs : s ⊆ e.source) (ht : ↑e '' s = t) : ∀ (a : ↑s), (PartialHomeomorph.homeomorphOfImageSubsetSource e hs ht) a = Set.MapsTo.restrict (↑e) s t (_ : Set.MapsTo (↑e) s t) a

@[simp]

theorem PartialHomeomorph.homeomorphOfImageSubsetSource_symm_apply {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] (e : PartialHomeomorph α β) {s : Set α} {t : Set β} (hs : s ⊆ e.source) (ht : ↑e '' s = t) : ∀ (a : ↑t), (Homeomorph.symm (PartialHomeomorph.homeomorphOfImageSubsetSource e hs ht)) a = Set.MapsTo.restrict (↑(PartialHomeomorph.symm e)) t s (_ : Set.MapsTo (↑(PartialHomeomorph.symm e)) t s) a

def PartialHomeomorph.homeomorphOfImageSubsetSource {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] (e : PartialHomeomorph α β) {s : Set α} {t : Set β} (hs : s ⊆ e.source) (ht : ↑e '' s = t) : ↑s ≃ₜ ↑t

The homeomorphism obtained by restricting a PartialHomeomorph to a subset of the source.

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@[simp]

theorem PartialHomeomorph.toHomeomorphSourceTarget_symm_apply_coe {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] (e : PartialHomeomorph α β) : ∀ (a : ↑e.target), ↑((Homeomorph.symm (PartialHomeomorph.toHomeomorphSourceTarget e)) a) = ↑(PartialHomeomorph.symm e) ↑a

@[simp]

theorem PartialHomeomorph.toHomeomorphSourceTarget_apply_coe {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] (e : PartialHomeomorph α β) : ∀ (a : ↑e.source), ↑((PartialHomeomorph.toHomeomorphSourceTarget e) a) = ↑e ↑a

def PartialHomeomorph.toHomeomorphSourceTarget {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] (e : PartialHomeomorph α β) : ↑e.source ≃ₜ ↑e.target

A partial homeomorphism defines a homeomorphism between its source and target.

Equations

• PartialHomeomorph.toHomeomorphSourceTarget e = PartialHomeomorph.homeomorphOfImageSubsetSource e (_ : e.source ⊆ e.source) (_ : ↑e '' e.source = e.target)

theorem PartialHomeomorph.secondCountableTopology_source {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] [SecondCountableTopology β] (e : PartialHomeomorph α β) : SecondCountableTopology ↑e.source

theorem PartialHomeomorph.nhds_eq_comap_inf_principal {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] (e : PartialHomeomorph α β) {x : α} (hx : x ∈ e.source) : nhds x = Filter.comap (↑e) (nhds (↑e x)) ⊓ Filter.principal e.source

@[simp]

theorem PartialHomeomorph.toHomeomorphOfSourceEqUnivTargetEqUniv_symm_apply {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] (e : PartialHomeomorph α β) (h : e.source = Set.univ) (h' : e.target = Set.univ) : ⇑(Homeomorph.symm (PartialHomeomorph.toHomeomorphOfSourceEqUnivTargetEqUniv e h h')) = ↑(PartialHomeomorph.symm e)

@[simp]

theorem PartialHomeomorph.toHomeomorphOfSourceEqUnivTargetEqUniv_apply {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] (e : PartialHomeomorph α β) (h : e.source = Set.univ) (h' : e.target = Set.univ) : ⇑(PartialHomeomorph.toHomeomorphOfSourceEqUnivTargetEqUniv e h h') = ↑e

def PartialHomeomorph.toHomeomorphOfSourceEqUnivTargetEqUniv {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] (e : PartialHomeomorph α β) (h : e.source = Set.univ) (h' : e.target = Set.univ) : α ≃ₜ β

If a partial homeomorphism has source and target equal to univ, then it induces a homeomorphism between the whole spaces, expressed in this definition.

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• One or more equations did not get rendered due to their size.

theorem PartialHomeomorph.openEmbedding_restrict {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] (e : PartialHomeomorph α β) : OpenEmbedding (Set.restrict e.source ↑e)

theorem PartialHomeomorph.to_openEmbedding {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] (e : PartialHomeomorph α β) (h : e.source = Set.univ) : OpenEmbedding ↑e

A partial homeomorphism whose source is all of α defines an open embedding of α into β. The converse is also true; see OpenEmbedding.toPartialHomeomorph.

@[simp]

theorem Homeomorph.refl_toPartialHomeomorph {α : Type u_1} [TopologicalSpace α] : Homeomorph.toPartialHomeomorph (Homeomorph.refl α) = PartialHomeomorph.refl α

@[simp]

theorem Homeomorph.symm_toPartialHomeomorph {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] (e : α ≃ₜ β) : Homeomorph.toPartialHomeomorph (Homeomorph.symm e) = PartialHomeomorph.symm (Homeomorph.toPartialHomeomorph e)

@[simp]

theorem Homeomorph.trans_toPartialHomeomorph {α : Type u_1} {β : Type u_2} {γ : Type u_3} [TopologicalSpace α] [TopologicalSpace β] [TopologicalSpace γ] (e : α ≃ₜ β) (e' : β ≃ₜ γ) : Homeomorph.toPartialHomeomorph (Homeomorph.trans e e') = PartialHomeomorph.trans (Homeomorph.toPartialHomeomorph e) (Homeomorph.toPartialHomeomorph e')

@[simp]

theorem Homeomorph.transPartialHomeomorph_apply {α : Type u_1} {β : Type u_2} {γ : Type u_3} [TopologicalSpace α] [TopologicalSpace β] [TopologicalSpace γ] (e : α ≃ₜ β) (f' : PartialHomeomorph β γ) : ↑(Homeomorph.transPartialHomeomorph e f') = ↑f' ∘ ⇑e

@[simp]

theorem Homeomorph.transPartialHomeomorph_symm_apply {α : Type u_1} {β : Type u_2} {γ : Type u_3} [TopologicalSpace α] [TopologicalSpace β] [TopologicalSpace γ] (e : α ≃ₜ β) (f' : PartialHomeomorph β γ) : ↑(PartialHomeomorph.symm (Homeomorph.transPartialHomeomorph e f')) = ⇑(Homeomorph.symm e) ∘ ↑(PartialHomeomorph.symm f')

@[simp]

theorem Homeomorph.transPartialHomeomorph_source {α : Type u_1} {β : Type u_2} {γ : Type u_3} [TopologicalSpace α] [TopologicalSpace β] [TopologicalSpace γ] (e : α ≃ₜ β) (f' : PartialHomeomorph β γ) : (Homeomorph.transPartialHomeomorph e f').toPartialEquiv.source = ⇑e ⁻¹' f'.source

@[simp]

theorem Homeomorph.transPartialHomeomorph_target {α : Type u_1} {β : Type u_2} {γ : Type u_3} [TopologicalSpace α] [TopologicalSpace β] [TopologicalSpace γ] (e : α ≃ₜ β) (f' : PartialHomeomorph β γ) : (Homeomorph.transPartialHomeomorph e f').toPartialEquiv.target = f'.target

def Homeomorph.transPartialHomeomorph {α : Type u_1} {β : Type u_2} {γ : Type u_3} [TopologicalSpace α] [TopologicalSpace β] [TopologicalSpace γ] (e : α ≃ₜ β) (f' : PartialHomeomorph β γ) : PartialHomeomorph α γ

Precompose a partial homeomorphism with a homeomorphism. We modify the source and target to have better definitional behavior.

Equations

• One or more equations did not get rendered due to their size.

theorem Homeomorph.transPartialHomeomorph_eq_trans {α : Type u_1} {β : Type u_2} {γ : Type u_3} [TopologicalSpace α] [TopologicalSpace β] [TopologicalSpace γ] (e : α ≃ₜ β) (f' : PartialHomeomorph β γ) : Homeomorph.transPartialHomeomorph e f' = PartialHomeomorph.trans (Homeomorph.toPartialHomeomorph e) f'

@[simp]

theorem Homeomorph.transPartialHomeomorph_trans {α : Type u_1} {β : Type u_2} {γ : Type u_3} {δ : Type u_4} [TopologicalSpace α] [TopologicalSpace β] [TopologicalSpace γ] [TopologicalSpace δ] (e : α ≃ₜ β) (f : PartialHomeomorph β γ) (f' : PartialHomeomorph γ δ) : PartialHomeomorph.trans (Homeomorph.transPartialHomeomorph e f) f' = Homeomorph.transPartialHomeomorph e (PartialHomeomorph.trans f f')

@[simp]

theorem Homeomorph.trans_transPartialHomeomorph {α : Type u_1} {β : Type u_2} {γ : Type u_3} {δ : Type u_4} [TopologicalSpace α] [TopologicalSpace β] [TopologicalSpace γ] [TopologicalSpace δ] (e : α ≃ₜ β) (e' : β ≃ₜ γ) (f'' : PartialHomeomorph γ δ) : Homeomorph.transPartialHomeomorph (Homeomorph.trans e e') f'' = Homeomorph.transPartialHomeomorph e (Homeomorph.transPartialHomeomorph e' f'')

@[simp]

theorem OpenEmbedding.toPartialHomeomorph_target {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] (f : α → β) (h : OpenEmbedding f) [Nonempty α] : (OpenEmbedding.toPartialHomeomorph f h).toPartialEquiv.target = Set.range f

@[simp]

theorem OpenEmbedding.toPartialHomeomorph_apply {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] (f : α → β) (h : OpenEmbedding f) [Nonempty α] : ↑(OpenEmbedding.toPartialHomeomorph f h) = f

@[simp]

theorem OpenEmbedding.toPartialHomeomorph_source {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] (f : α → β) (h : OpenEmbedding f) [Nonempty α] : (OpenEmbedding.toPartialHomeomorph f h).toPartialEquiv.source = Set.univ

noncomputable def OpenEmbedding.toPartialHomeomorph {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] (f : α → β) (h : OpenEmbedding f) [Nonempty α] : PartialHomeomorph α β

An open embedding of α into β, with α nonempty, defines a partial homeomorphism whose source is all of α. The converse is also true; see PartialHomeomorph.to_openEmbedding.

Equations

• One or more equations did not get rendered due to their size.

theorem OpenEmbedding.toPartialHomeomorph_left_inv {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] (f : α → β) (h : OpenEmbedding f) [Nonempty α] {x : α} : ↑(PartialHomeomorph.symm (OpenEmbedding.toPartialHomeomorph f h)) (f x) = x

theorem OpenEmbedding.toPartialHomeomorph_right_inv {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] (f : α → β) (h : OpenEmbedding f) [Nonempty α] {x : β} (hx : x ∈ Set.range f) : f (↑(PartialHomeomorph.symm (OpenEmbedding.toPartialHomeomorph f h)) x) = x

noncomputable def TopologicalSpace.Opens.partialHomeomorphSubtypeCoe {α : Type u_1} [TopologicalSpace α] (s : TopologicalSpace.Opens α) [Nonempty ↥s] : PartialHomeomorph (↥s) α

The inclusion of an open subset s of a space α into α is a partial homeomorphism from the subtype s to α.

Equations

• TopologicalSpace.Opens.partialHomeomorphSubtypeCoe s = OpenEmbedding.toPartialHomeomorph Subtype.val (_ : OpenEmbedding Subtype.val)

@[simp]

theorem TopologicalSpace.Opens.partialHomeomorphSubtypeCoe_coe {α : Type u_1} [TopologicalSpace α] (s : TopologicalSpace.Opens α) [Nonempty ↥s] : ↑(TopologicalSpace.Opens.partialHomeomorphSubtypeCoe s) = Subtype.val

@[simp]

theorem TopologicalSpace.Opens.partialHomeomorphSubtypeCoe_source {α : Type u_1} [TopologicalSpace α] (s : TopologicalSpace.Opens α) [Nonempty ↥s] : (TopologicalSpace.Opens.partialHomeomorphSubtypeCoe s).toPartialEquiv.source = Set.univ

@[simp]

theorem TopologicalSpace.Opens.partialHomeomorphSubtypeCoe_target {α : Type u_1} [TopologicalSpace α] (s : TopologicalSpace.Opens α) [Nonempty ↥s] : (TopologicalSpace.Opens.partialHomeomorphSubtypeCoe s).toPartialEquiv.target = ↑s

@[simp]

theorem PartialHomeomorph.transHomeomorph_apply {α : Type u_1} {β : Type u_2} {γ : Type u_3} [TopologicalSpace α] [TopologicalSpace β] [TopologicalSpace γ] (e : PartialHomeomorph α β) (f' : β ≃ₜ γ) : ↑(PartialHomeomorph.transHomeomorph e f') = ⇑f' ∘ ↑e

@[simp]

theorem PartialHomeomorph.transHomeomorph_symm_apply {α : Type u_1} {β : Type u_2} {γ : Type u_3} [TopologicalSpace α] [TopologicalSpace β] [TopologicalSpace γ] (e : PartialHomeomorph α β) (f' : β ≃ₜ γ) : ↑(PartialHomeomorph.symm (PartialHomeomorph.transHomeomorph e f')) = ↑(PartialHomeomorph.symm e) ∘ ⇑(Homeomorph.symm f')

@[simp]

theorem PartialHomeomorph.transHomeomorph_source {α : Type u_1} {β : Type u_2} {γ : Type u_3} [TopologicalSpace α] [TopologicalSpace β] [TopologicalSpace γ] (e : PartialHomeomorph α β) (f' : β ≃ₜ γ) : (PartialHomeomorph.transHomeomorph e f').toPartialEquiv.source = e.source

@[simp]

theorem PartialHomeomorph.transHomeomorph_target {α : Type u_1} {β : Type u_2} {γ : Type u_3} [TopologicalSpace α] [TopologicalSpace β] [TopologicalSpace γ] (e : PartialHomeomorph α β) (f' : β ≃ₜ γ) : (PartialHomeomorph.transHomeomorph e f').toPartialEquiv.target = ⇑(Homeomorph.symm f') ⁻¹' e.target

def PartialHomeomorph.transHomeomorph {α : Type u_1} {β : Type u_2} {γ : Type u_3} [TopologicalSpace α] [TopologicalSpace β] [TopologicalSpace γ] (e : PartialHomeomorph α β) (f' : β ≃ₜ γ) : PartialHomeomorph α γ

Postcompose a partial homeomorphism with a homeomorphism. We modify the source and target to have better definitional behavior.

Equations

• One or more equations did not get rendered due to their size.

theorem PartialHomeomorph.transHomeomorph_eq_trans {α : Type u_1} {β : Type u_2} {γ : Type u_3} [TopologicalSpace α] [TopologicalSpace β] [TopologicalSpace γ] (e : PartialHomeomorph α β) (f' : β ≃ₜ γ) : PartialHomeomorph.transHomeomorph e f' = PartialHomeomorph.trans e (Homeomorph.toPartialHomeomorph f')

@[simp]

theorem PartialHomeomorph.transHomeomorph_transHomeomorph {α : Type u_1} {β : Type u_2} {γ : Type u_3} {δ : Type u_4} [TopologicalSpace α] [TopologicalSpace β] [TopologicalSpace γ] [TopologicalSpace δ] (e : PartialHomeomorph α β) (f' : β ≃ₜ γ) (f'' : γ ≃ₜ δ) : PartialHomeomorph.transHomeomorph (PartialHomeomorph.transHomeomorph e f') f'' = PartialHomeomorph.transHomeomorph e (Homeomorph.trans f' f'')

@[simp]

theorem PartialHomeomorph.trans_transHomeomorph {α : Type u_1} {β : Type u_2} {γ : Type u_3} {δ : Type u_4} [TopologicalSpace α] [TopologicalSpace β] [TopologicalSpace γ] [TopologicalSpace δ] (e : PartialHomeomorph α β) (e' : PartialHomeomorph β γ) (f'' : γ ≃ₜ δ) : PartialHomeomorph.transHomeomorph (PartialHomeomorph.trans e e') f'' = PartialHomeomorph.trans e (PartialHomeomorph.transHomeomorph e' f'')

noncomputable def PartialHomeomorph.subtypeRestr {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] (e : PartialHomeomorph α β) (s : TopologicalSpace.Opens α) [Nonempty ↥s] : PartialHomeomorph (↥s) β

The restriction of a partial homeomorphism e to an open subset s of the domain type produces a partial homeomorphism whose domain is the subtype s.

Equations

• PartialHomeomorph.subtypeRestr e s = PartialHomeomorph.trans (TopologicalSpace.Opens.partialHomeomorphSubtypeCoe s) e

theorem PartialHomeomorph.subtypeRestr_def {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] (e : PartialHomeomorph α β) (s : TopologicalSpace.Opens α) [Nonempty ↥s] : PartialHomeomorph.subtypeRestr e s = PartialHomeomorph.trans (TopologicalSpace.Opens.partialHomeomorphSubtypeCoe s) e

@[simp]

theorem PartialHomeomorph.subtypeRestr_coe {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] (e : PartialHomeomorph α β) (s : TopologicalSpace.Opens α) [Nonempty ↥s] : ↑(PartialHomeomorph.subtypeRestr e s) = Set.restrict ↑s ↑e

@[simp]

theorem PartialHomeomorph.subtypeRestr_source {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] (e : PartialHomeomorph α β) (s : TopologicalSpace.Opens α) [Nonempty ↥s] : (PartialHomeomorph.subtypeRestr e s).toPartialEquiv.source = Subtype.val ⁻¹' e.source

theorem PartialHomeomorph.map_subtype_source {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] (e : PartialHomeomorph α β) {s : TopologicalSpace.Opens α} [Nonempty ↥s] {x : ↥s} (hxe : ↑x ∈ e.source) : ↑e ↑x ∈ (PartialHomeomorph.subtypeRestr e s).toPartialEquiv.target

theorem PartialHomeomorph.subtypeRestr_symm_trans_subtypeRestr {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] (s : TopologicalSpace.Opens α) [Nonempty ↥s] (f : PartialHomeomorph α β) (f' : PartialHomeomorph α β) : PartialHomeomorph.trans (PartialHomeomorph.symm (PartialHomeomorph.subtypeRestr f s)) (PartialHomeomorph.subtypeRestr f' s) ≈ PartialHomeomorph.restr (PartialHomeomorph.trans (PartialHomeomorph.symm f) f') (f.target ∩ ↑(PartialHomeomorph.symm f) ⁻¹' ↑s)

theorem PartialHomeomorph.subtypeRestr_symm_eqOn {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] (e : PartialHomeomorph α β) (U : TopologicalSpace.Opens α) [Nonempty ↥U] : Set.EqOn (↑(PartialHomeomorph.symm e)) (Subtype.val ∘ ↑(PartialHomeomorph.symm (PartialHomeomorph.subtypeRestr e U))) (PartialHomeomorph.subtypeRestr e U).toPartialEquiv.target

theorem PartialHomeomorph.subtypeRestr_symm_eqOn_of_le {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] (e : PartialHomeomorph α β) {U : TopologicalSpace.Opens α} {V : TopologicalSpace.Opens α} [Nonempty ↥U] [Nonempty ↥V] (hUV : U ≤ V) : Set.EqOn (↑(PartialHomeomorph.symm (PartialHomeomorph.subtypeRestr e V))) (Set.inclusion hUV ∘ ↑(PartialHomeomorph.symm (PartialHomeomorph.subtypeRestr e U))) (PartialHomeomorph.subtypeRestr e U).toPartialEquiv.target