Partial homeomorphisms #

This file defines homeomorphisms between open subsets of topological spaces. An element e of PartialHomeomorph X Y is an extension of PartialEquiv X Y, 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 (X : Type u_7) (Y : Type u_8) [] [] extends :
Type (max u_7 u_8)

Partial homeomorphisms, defined on open subsets of the space

• toFun : XY
• invFun : YX
• source : Set X
• target : Set Y
• map_source' : ∀ ⦃x : X⦄, x self.sourceself.toPartialEquiv x self.target
• map_target' : ∀ ⦃x : Y⦄, x self.targetself.invFun x self.source
• left_inv' : ∀ ⦃x : X⦄, x self.sourceself.invFun (self.toPartialEquiv x) = x
• right_inv' : ∀ ⦃x : Y⦄, x self.targetself.toPartialEquiv (self.invFun x) = x
• open_source : IsOpen self.source
• open_target : IsOpen self.target
• continuousOn_toFun : ContinuousOn (↑self.toPartialEquiv) self.source
• continuousOn_invFun : ContinuousOn self.invFun self.target
Instances For
theorem PartialHomeomorph.open_source {X : Type u_7} {Y : Type u_8} [] [] (self : ) :
IsOpen self.source
theorem PartialHomeomorph.open_target {X : Type u_7} {Y : Type u_8} [] [] (self : ) :
IsOpen self.target
theorem PartialHomeomorph.continuousOn_toFun {X : Type u_7} {Y : Type u_8} [] [] (self : ) :
ContinuousOn (↑self.toPartialEquiv) self.source
theorem PartialHomeomorph.continuousOn_invFun {X : Type u_7} {Y : Type u_8} [] [] (self : ) :
ContinuousOn self.invFun self.target

Basic properties; inverse (symm instance)

def PartialHomeomorph.toFun' {X : Type u_1} {Y : Type u_3} [] [] (e : ) :
XY

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
Instances For
instance PartialHomeomorph.instCoeFunForall {X : Type u_1} {Y : Type u_3} [] [] :
CoeFun fun (x : ) => XY

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

Equations
• PartialHomeomorph.instCoeFunForall = { coe := fun (e : ) => e }
def PartialHomeomorph.symm {X : Type u_1} {Y : Type u_3} [] [] (e : ) :

The inverse of a partial homeomorphism

Equations
• e.symm = { toPartialEquiv := e.symm, open_source := , open_target := , continuousOn_toFun := , continuousOn_invFun := }
Instances For
def PartialHomeomorph.Simps.apply {X : Type u_1} {Y : Type u_3} [] [] (e : ) :
XY

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

Equations
Instances For
def PartialHomeomorph.Simps.symm_apply {X : Type u_1} {Y : Type u_3} [] [] (e : ) :
YX

See Note [custom simps projection]

Equations
• = e.symm
Instances For
theorem PartialHomeomorph.continuousOn {X : Type u_1} {Y : Type u_3} [] [] (e : ) :
ContinuousOn (↑e) e.source
theorem PartialHomeomorph.continuousOn_symm {X : Type u_1} {Y : Type u_3} [] [] (e : ) :
ContinuousOn (↑e.symm) e.target
@[simp]
theorem PartialHomeomorph.mk_coe {X : Type u_1} {Y : Type u_3} [] [] (e : ) (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 {X : Type u_1} {Y : Type u_3} [] [] (e : ) (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 }.symm = e.symm
theorem PartialHomeomorph.toPartialEquiv_injective {X : Type u_1} {Y : Type u_3} [] [] :
Function.Injective PartialHomeomorph.toPartialEquiv
@[simp]
theorem PartialHomeomorph.toFun_eq_coe {X : Type u_1} {Y : Type u_3} [] [] (e : ) :
e.toPartialEquiv = e
@[simp]
theorem PartialHomeomorph.invFun_eq_coe {X : Type u_1} {Y : Type u_3} [] [] (e : ) :
e.invFun = e.symm
@[simp]
theorem PartialHomeomorph.coe_coe {X : Type u_1} {Y : Type u_3} [] [] (e : ) :
e.toPartialEquiv = e
@[simp]
theorem PartialHomeomorph.coe_coe_symm {X : Type u_1} {Y : Type u_3} [] [] (e : ) :
e.symm = e.symm
@[simp]
theorem PartialHomeomorph.map_source {X : Type u_1} {Y : Type u_3} [] [] (e : ) {x : X} (h : x e.source) :
e x e.target
theorem PartialHomeomorph.map_source'' {X : Type u_1} {Y : Type u_3} [] [] (e : ) :
e '' e.source e.target

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

@[simp]
theorem PartialHomeomorph.map_target {X : Type u_1} {Y : Type u_3} [] [] (e : ) {x : Y} (h : x e.target) :
e.symm x e.source
@[simp]
theorem PartialHomeomorph.left_inv {X : Type u_1} {Y : Type u_3} [] [] (e : ) {x : X} (h : x e.source) :
e.symm (e x) = x
@[simp]
theorem PartialHomeomorph.right_inv {X : Type u_1} {Y : Type u_3} [] [] (e : ) {x : Y} (h : x e.target) :
e (e.symm x) = x
theorem PartialHomeomorph.eq_symm_apply {X : Type u_1} {Y : Type u_3} [] [] (e : ) {x : X} {y : Y} (hx : x e.source) (hy : y e.target) :
x = e.symm y e x = y
theorem PartialHomeomorph.mapsTo {X : Type u_1} {Y : Type u_3} [] [] (e : ) :
Set.MapsTo (↑e) e.source e.target
theorem PartialHomeomorph.symm_mapsTo {X : Type u_1} {Y : Type u_3} [] [] (e : ) :
Set.MapsTo (↑e.symm) e.target e.source
theorem PartialHomeomorph.leftInvOn {X : Type u_1} {Y : Type u_3} [] [] (e : ) :
Set.LeftInvOn (↑e.symm) (↑e) e.source
theorem PartialHomeomorph.rightInvOn {X : Type u_1} {Y : Type u_3} [] [] (e : ) :
Set.RightInvOn (↑e.symm) (↑e) e.target
theorem PartialHomeomorph.invOn {X : Type u_1} {Y : Type u_3} [] [] (e : ) :
Set.InvOn (↑e.symm) (↑e) e.source e.target
theorem PartialHomeomorph.injOn {X : Type u_1} {Y : Type u_3} [] [] (e : ) :
Set.InjOn (↑e) e.source
theorem PartialHomeomorph.bijOn {X : Type u_1} {Y : Type u_3} [] [] (e : ) :
Set.BijOn (↑e) e.source e.target
theorem PartialHomeomorph.surjOn {X : Type u_1} {Y : Type u_3} [] [] (e : ) :
Set.SurjOn (↑e) e.source e.target
@[simp]
theorem Homeomorph.toPartialHomeomorphOfImageEq_symm_apply {X : Type u_1} {Y : Type u_3} [] [] (e : X ≃ₜ Y) (s : Set X) (hs : ) (t : Set Y) (h : e '' s = t) :
(e.toPartialHomeomorphOfImageEq s hs t h).symm = e.symm
@[simp]
theorem Homeomorph.toPartialHomeomorphOfImageEq_toPartialEquiv {X : Type u_1} {Y : Type u_3} [] [] (e : X ≃ₜ Y) (s : Set X) (hs : ) (t : Set Y) (h : e '' s = t) :
(e.toPartialHomeomorphOfImageEq s hs t h).toPartialEquiv = e.toPartialEquivOfImageEq s t h
@[simp]
theorem Homeomorph.toPartialHomeomorphOfImageEq_apply {X : Type u_1} {Y : Type u_3} [] [] (e : X ≃ₜ Y) (s : Set X) (hs : ) (t : Set Y) (h : e '' s = t) :
(e.toPartialHomeomorphOfImageEq s hs t h) = e
theorem Homeomorph.toPartialHomeomorphOfImageEq_source {X : Type u_1} {Y : Type u_3} [] [] (e : X ≃ₜ Y) (s : Set X) (hs : ) (t : Set Y) (h : e '' s = t) :
(e.toPartialHomeomorphOfImageEq s hs t h).source = s
theorem Homeomorph.toPartialHomeomorphOfImageEq_target {X : Type u_1} {Y : Type u_3} [] [] (e : X ≃ₜ Y) (s : Set X) (hs : ) (t : Set Y) (h : e '' s = t) :
(e.toPartialHomeomorphOfImageEq s hs t h).target = t
def Homeomorph.toPartialHomeomorphOfImageEq {X : Type u_1} {Y : Type u_3} [] [] (e : X ≃ₜ Y) (s : Set X) (hs : ) (t : Set Y) (h : e '' s = t) :

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

Equations
• e.toPartialHomeomorphOfImageEq s hs t h = { toPartialEquiv := e.toPartialEquivOfImageEq s t h, open_source := hs, open_target := , continuousOn_toFun := , continuousOn_invFun := }
Instances For
@[simp]
theorem Homeomorph.toPartialHomeomorph_source {X : Type u_1} {Y : Type u_3} [] [] (e : X ≃ₜ Y) :
e.toPartialHomeomorph.source = Set.univ
@[simp]
theorem Homeomorph.toPartialHomeomorph_target {X : Type u_1} {Y : Type u_3} [] [] (e : X ≃ₜ Y) :
e.toPartialHomeomorph.target = Set.univ
@[simp]
theorem Homeomorph.toPartialHomeomorph_apply {X : Type u_1} {Y : Type u_3} [] [] (e : X ≃ₜ Y) :
e.toPartialHomeomorph = e
@[simp]
theorem Homeomorph.toPartialHomeomorph_symm_apply {X : Type u_1} {Y : Type u_3} [] [] (e : X ≃ₜ Y) :
e.toPartialHomeomorph.symm = e.symm
def Homeomorph.toPartialHomeomorph {X : Type u_1} {Y : Type u_3} [] [] (e : X ≃ₜ Y) :

A homeomorphism induces a partial homeomorphism on the whole space

Equations
• e.toPartialHomeomorph = e.toPartialHomeomorphOfImageEq Set.univ Set.univ
Instances For
def PartialHomeomorph.replaceEquiv {X : Type u_1} {Y : Type u_3} [] [] (e : ) (e' : ) (h : e.toPartialEquiv = e') :

Replace toPartialEquiv field to provide better definitional equalities.

Equations
• e.replaceEquiv e' h = { toPartialEquiv := e', open_source := , open_target := , continuousOn_toFun := , continuousOn_invFun := }
Instances For
theorem PartialHomeomorph.replaceEquiv_eq_self {X : Type u_1} {Y : Type u_3} [] [] (e : ) (e' : ) (h : e.toPartialEquiv = e') :
e.replaceEquiv e' h = e
theorem PartialHomeomorph.source_preimage_target {X : Type u_1} {Y : Type u_3} [] [] (e : ) :
e.source e ⁻¹' e.target
theorem PartialHomeomorph.eventually_left_inverse {X : Type u_1} {Y : Type u_3} [] [] (e : ) {x : X} (hx : x e.source) :
∀ᶠ (y : X) in nhds x, e.symm (e y) = y
theorem PartialHomeomorph.eventually_left_inverse' {X : Type u_1} {Y : Type u_3} [] [] (e : ) {x : Y} (hx : x e.target) :
∀ᶠ (y : X) in nhds (e.symm x), e.symm (e y) = y
theorem PartialHomeomorph.eventually_right_inverse {X : Type u_1} {Y : Type u_3} [] [] (e : ) {x : Y} (hx : x e.target) :
∀ᶠ (y : Y) in nhds x, e (e.symm y) = y
theorem PartialHomeomorph.eventually_right_inverse' {X : Type u_1} {Y : Type u_3} [] [] (e : ) {x : X} (hx : x e.source) :
∀ᶠ (y : Y) in nhds (e x), e (e.symm y) = y
theorem PartialHomeomorph.eventually_ne_nhdsWithin {X : Type u_1} {Y : Type u_3} [] [] (e : ) {x : X} (hx : x e.source) :
∀ᶠ (x' : X) in nhdsWithin x {x}, e x' e x
theorem PartialHomeomorph.nhdsWithin_source_inter {X : Type u_1} {Y : Type u_3} [] [] (e : ) {x : X} (hx : x e.source) (s : Set X) :
nhdsWithin x (e.source s) =
theorem PartialHomeomorph.nhdsWithin_target_inter {X : Type u_1} {Y : Type u_3} [] [] (e : ) {x : Y} (hx : x e.target) (s : Set Y) :
nhdsWithin x (e.target s) =
theorem PartialHomeomorph.image_eq_target_inter_inv_preimage {X : Type u_1} {Y : Type u_3} [] [] (e : ) {s : Set X} (h : s e.source) :
e '' s = e.target e.symm ⁻¹' s
theorem PartialHomeomorph.image_source_inter_eq' {X : Type u_1} {Y : Type u_3} [] [] (e : ) (s : Set X) :
e '' (e.source s) = e.target e.symm ⁻¹' s
theorem PartialHomeomorph.image_source_inter_eq {X : Type u_1} {Y : Type u_3} [] [] (e : ) (s : Set X) :
e '' (e.source s) = e.target e.symm ⁻¹' (e.source s)
theorem PartialHomeomorph.symm_image_eq_source_inter_preimage {X : Type u_1} {Y : Type u_3} [] [] (e : ) {s : Set Y} (h : s e.target) :
e.symm '' s = e.source e ⁻¹' s
theorem PartialHomeomorph.symm_image_target_inter_eq {X : Type u_1} {Y : Type u_3} [] [] (e : ) (s : Set Y) :
e.symm '' (e.target s) = e.source e ⁻¹' (e.target s)
theorem PartialHomeomorph.source_inter_preimage_inv_preimage {X : Type u_1} {Y : Type u_3} [] [] (e : ) (s : Set X) :
e.source e ⁻¹' (e.symm ⁻¹' s) = e.source s
theorem PartialHomeomorph.target_inter_inv_preimage_preimage {X : Type u_1} {Y : Type u_3} [] [] (e : ) (s : Set Y) :
e.target e.symm ⁻¹' (e ⁻¹' s) = e.target s
theorem PartialHomeomorph.source_inter_preimage_target_inter {X : Type u_1} {Y : Type u_3} [] [] (e : ) (s : Set Y) :
e.source e ⁻¹' (e.target s) = e.source e ⁻¹' s
theorem PartialHomeomorph.image_source_eq_target {X : Type u_1} {Y : Type u_3} [] [] (e : ) :
e '' e.source = e.target
theorem PartialHomeomorph.symm_image_target_eq_source {X : Type u_1} {Y : Type u_3} [] [] (e : ) :
e.symm '' e.target = e.source
theorem PartialHomeomorph.ext_iff {X : Type u_1} {Y : Type u_3} [] [] {e : } {e' : } :
e = e' (∀ (x : X), e x = e' x) (∀ (x : Y), e.symm x = e'.symm x) e.source = e'.source
theorem PartialHomeomorph.ext {X : Type u_1} {Y : Type u_3} [] [] (e : ) (e' : ) (h : ∀ (x : X), e x = e' x) (hinv : ∀ (x : Y), e.symm x = e'.symm 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.

@[simp]
theorem PartialHomeomorph.symm_toPartialEquiv {X : Type u_1} {Y : Type u_3} [] [] (e : ) :
e.symm.toPartialEquiv = e.symm
theorem PartialHomeomorph.symm_source {X : Type u_1} {Y : Type u_3} [] [] (e : ) :
e.symm.source = e.target
theorem PartialHomeomorph.symm_target {X : Type u_1} {Y : Type u_3} [] [] (e : ) :
e.symm.target = e.source
@[simp]
theorem PartialHomeomorph.symm_symm {X : Type u_1} {Y : Type u_3} [] [] (e : ) :
e.symm.symm = e
theorem PartialHomeomorph.symm_bijective {X : Type u_1} {Y : Type u_3} [] [] :
Function.Bijective PartialHomeomorph.symm
theorem PartialHomeomorph.continuousAt {X : Type u_1} {Y : Type u_3} [] [] (e : ) {x : X} (h : x e.source) :
ContinuousAt (↑e) x

A partial homeomorphism is continuous at any point of its source

theorem PartialHomeomorph.continuousAt_symm {X : Type u_1} {Y : Type u_3} [] [] (e : ) {x : Y} (h : x e.target) :
ContinuousAt (↑e.symm) x

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

theorem PartialHomeomorph.tendsto_symm {X : Type u_1} {Y : Type u_3} [] [] (e : ) {x : X} (hx : x e.source) :
Filter.Tendsto (↑e.symm) (nhds (e x)) (nhds x)
theorem PartialHomeomorph.map_nhds_eq {X : Type u_1} {Y : Type u_3} [] [] (e : ) {x : X} (hx : x e.source) :
Filter.map (↑e) (nhds x) = nhds (e x)
theorem PartialHomeomorph.symm_map_nhds_eq {X : Type u_1} {Y : Type u_3} [] [] (e : ) {x : X} (hx : x e.source) :
Filter.map (↑e.symm) (nhds (e x)) = nhds x
theorem PartialHomeomorph.image_mem_nhds {X : Type u_1} {Y : Type u_3} [] [] (e : ) {x : X} (hx : x e.source) {s : Set X} (hs : s nhds x) :
e '' s nhds (e x)
theorem PartialHomeomorph.map_nhdsWithin_eq {X : Type u_1} {Y : Type u_3} [] [] (e : ) {x : X} (hx : x e.source) (s : Set X) :
Filter.map (↑e) (nhdsWithin x s) = nhdsWithin (e x) (e '' (e.source s))
theorem PartialHomeomorph.map_nhdsWithin_preimage_eq {X : Type u_1} {Y : Type u_3} [] [] (e : ) {x : X} (hx : x e.source) (s : Set Y) :
Filter.map (↑e) (nhdsWithin x (e ⁻¹' s)) = nhdsWithin (e x) s
theorem PartialHomeomorph.eventually_nhds {X : Type u_1} {Y : Type u_3} [] [] (e : ) {x : X} (p : YProp) (hx : x e.source) :
(∀ᶠ (y : Y) in nhds (e x), p y) ∀ᶠ (x : X) in nhds x, p (e x)
theorem PartialHomeomorph.eventually_nhds' {X : Type u_1} {Y : Type u_3} [] [] (e : ) {x : X} (p : XProp) (hx : x e.source) :
(∀ᶠ (y : Y) in nhds (e x), p (e.symm y)) ∀ᶠ (x : X) in nhds x, p x
theorem PartialHomeomorph.eventually_nhdsWithin {X : Type u_1} {Y : Type u_3} [] [] (e : ) {x : X} (p : YProp) {s : Set X} (hx : x e.source) :
(∀ᶠ (y : Y) in nhdsWithin (e x) (e.symm ⁻¹' s), p y) ∀ᶠ (x : X) in , p (e x)
theorem PartialHomeomorph.eventually_nhdsWithin' {X : Type u_1} {Y : Type u_3} [] [] (e : ) {x : X} (p : XProp) {s : Set X} (hx : x e.source) :
(∀ᶠ (y : Y) in nhdsWithin (e x) (e.symm ⁻¹' s), p (e.symm y)) ∀ᶠ (x : X) in , p x
theorem PartialHomeomorph.preimage_eventuallyEq_target_inter_preimage_inter {X : Type u_1} {Y : Type u_3} {Z : Type u_5} [] [] [] {e : } {s : Set X} {t : Set Z} {x : X} {f : XZ} (hf : ) (hxe : x e.source) (ht : t nhds (f x)) :
e.symm ⁻¹' s =ᶠ[nhds (e x)] e.target e.symm ⁻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 Z).

theorem PartialHomeomorph.isOpen_inter_preimage {X : Type u_1} {Y : Type u_3} [] [] (e : ) {s : Set Y} (hs : ) :
IsOpen (e.source e ⁻¹' s)
theorem PartialHomeomorph.isOpen_inter_preimage_symm {X : Type u_1} {Y : Type u_3} [] [] (e : ) {s : Set X} (hs : ) :
IsOpen (e.target e.symm ⁻¹' s)
theorem PartialHomeomorph.isOpen_image_of_subset_source {X : Type u_1} {Y : Type u_3} [] [] (e : ) {s : Set X} (hs : ) (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 {X : Type u_1} {Y : Type u_3} [] [] (e : ) {s : Set X} (hs : ) :
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 {X : Type u_1} {Y : Type u_3} [] [] (e : ) {t : Set Y} (ht : ) (hte : t e.target) :
IsOpen (e.symm '' t)

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

theorem PartialHomeomorph.isOpen_symm_image_iff_of_subset_target {X : Type u_1} {Y : Type u_3} [] [] (e : ) {t : Set Y} (hs : t e.target) :
IsOpen (e.symm '' t)
theorem PartialHomeomorph.isOpen_image_iff_of_subset_source {X : Type u_1} {Y : Type u_3} [] [] (e : ) {s : Set X} (hs : s e.source) :
IsOpen (e '' s)

PartialHomeomorph.IsImage relation #

We say that t : Set Y is an image of s : Set X 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 {X : Type u_1} {Y : Type u_3} [] [] (e : ) (s : Set X) (t : Set Y) :

We say that t : Set Y is an image of s : Set X 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
• e.IsImage s t = ∀ ⦃x : X⦄, x e.source(e x t x s)
Instances For
theorem PartialHomeomorph.IsImage.toPartialEquiv {X : Type u_1} {Y : Type u_3} [] [] {e : } {s : Set X} {t : Set Y} (h : e.IsImage s t) :
e.IsImage s t
theorem PartialHomeomorph.IsImage.apply_mem_iff {X : Type u_1} {Y : Type u_3} [] [] {e : } {s : Set X} {t : Set Y} {x : X} (h : e.IsImage s t) (hx : x e.source) :
e x t x s
theorem PartialHomeomorph.IsImage.symm {X : Type u_1} {Y : Type u_3} [] [] {e : } {s : Set X} {t : Set Y} (h : e.IsImage s t) :
e.symm.IsImage t s
theorem PartialHomeomorph.IsImage.symm_apply_mem_iff {X : Type u_1} {Y : Type u_3} [] [] {e : } {s : Set X} {t : Set Y} {y : Y} (h : e.IsImage s t) (hy : y e.target) :
e.symm y s y t
@[simp]
theorem PartialHomeomorph.IsImage.symm_iff {X : Type u_1} {Y : Type u_3} [] [] {e : } {s : Set X} {t : Set Y} :
e.symm.IsImage t s e.IsImage s t
theorem PartialHomeomorph.IsImage.mapsTo {X : Type u_1} {Y : Type u_3} [] [] {e : } {s : Set X} {t : Set Y} (h : e.IsImage s t) :
Set.MapsTo (↑e) (e.source s) (e.target t)
theorem PartialHomeomorph.IsImage.symm_mapsTo {X : Type u_1} {Y : Type u_3} [] [] {e : } {s : Set X} {t : Set Y} (h : e.IsImage s t) :
Set.MapsTo (↑e.symm) (e.target t) (e.source s)
theorem PartialHomeomorph.IsImage.image_eq {X : Type u_1} {Y : Type u_3} [] [] {e : } {s : Set X} {t : Set Y} (h : e.IsImage s t) :
e '' (e.source s) = e.target t
theorem PartialHomeomorph.IsImage.symm_image_eq {X : Type u_1} {Y : Type u_3} [] [] {e : } {s : Set X} {t : Set Y} (h : e.IsImage s t) :
e.symm '' (e.target t) = e.source s
theorem PartialHomeomorph.IsImage.iff_preimage_eq {X : Type u_1} {Y : Type u_3} [] [] {e : } {s : Set X} {t : Set Y} :
e.IsImage s t e.source e ⁻¹' t = e.source s
theorem PartialHomeomorph.IsImage.of_preimage_eq {X : Type u_1} {Y : Type u_3} [] [] {e : } {s : Set X} {t : Set Y} :
e.source e ⁻¹' t = e.source se.IsImage s t

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

theorem PartialHomeomorph.IsImage.preimage_eq {X : Type u_1} {Y : Type u_3} [] [] {e : } {s : Set X} {t : Set Y} :
e.IsImage s te.source e ⁻¹' t = e.source s

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

theorem PartialHomeomorph.IsImage.iff_symm_preimage_eq {X : Type u_1} {Y : Type u_3} [] [] {e : } {s : Set X} {t : Set Y} :
e.IsImage s t e.target e.symm ⁻¹' s = e.target t
theorem PartialHomeomorph.IsImage.symm_preimage_eq {X : Type u_1} {Y : Type u_3} [] [] {e : } {s : Set X} {t : Set Y} :
e.IsImage s te.target e.symm ⁻¹' s = e.target t

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

theorem PartialHomeomorph.IsImage.of_symm_preimage_eq {X : Type u_1} {Y : Type u_3} [] [] {e : } {s : Set X} {t : Set Y} :
e.target e.symm ⁻¹' s = e.target te.IsImage s t

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

theorem PartialHomeomorph.IsImage.iff_symm_preimage_eq' {X : Type u_1} {Y : Type u_3} [] [] {e : } {s : Set X} {t : Set Y} :
e.IsImage s t e.target e.symm ⁻¹' (e.source s) = e.target t
theorem PartialHomeomorph.IsImage.symm_preimage_eq' {X : Type u_1} {Y : Type u_3} [] [] {e : } {s : Set X} {t : Set Y} :
e.IsImage s te.target e.symm ⁻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' {X : Type u_1} {Y : Type u_3} [] [] {e : } {s : Set X} {t : Set Y} :
e.target e.symm ⁻¹' (e.source s) = e.target te.IsImage s t

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

theorem PartialHomeomorph.IsImage.iff_preimage_eq' {X : Type u_1} {Y : Type u_3} [] [] {e : } {s : Set X} {t : Set Y} :
e.IsImage s t e.source e ⁻¹' (e.target t) = e.source s
theorem PartialHomeomorph.IsImage.of_preimage_eq' {X : Type u_1} {Y : Type u_3} [] [] {e : } {s : Set X} {t : Set Y} :
e.source e ⁻¹' (e.target t) = e.source se.IsImage s t

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

theorem PartialHomeomorph.IsImage.preimage_eq' {X : Type u_1} {Y : Type u_3} [] [] {e : } {s : Set X} {t : Set Y} :
e.IsImage s te.source e ⁻¹' (e.target t) = e.source s

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

theorem PartialHomeomorph.IsImage.of_image_eq {X : Type u_1} {Y : Type u_3} [] [] {e : } {s : Set X} {t : Set Y} (h : e '' (e.source s) = e.target t) :
e.IsImage s t
theorem PartialHomeomorph.IsImage.of_symm_image_eq {X : Type u_1} {Y : Type u_3} [] [] {e : } {s : Set X} {t : Set Y} (h : e.symm '' (e.target t) = e.source s) :
e.IsImage s t
theorem PartialHomeomorph.IsImage.compl {X : Type u_1} {Y : Type u_3} [] [] {e : } {s : Set X} {t : Set Y} (h : e.IsImage s t) :
e.IsImage s t
theorem PartialHomeomorph.IsImage.inter {X : Type u_1} {Y : Type u_3} [] [] {e : } {s : Set X} {t : Set Y} {s' : Set X} {t' : Set Y} (h : e.IsImage s t) (h' : e.IsImage s' t') :
e.IsImage (s s') (t t')
theorem PartialHomeomorph.IsImage.union {X : Type u_1} {Y : Type u_3} [] [] {e : } {s : Set X} {t : Set Y} {s' : Set X} {t' : Set Y} (h : e.IsImage s t) (h' : e.IsImage s' t') :
e.IsImage (s s') (t t')
theorem PartialHomeomorph.IsImage.diff {X : Type u_1} {Y : Type u_3} [] [] {e : } {s : Set X} {t : Set Y} {s' : Set X} {t' : Set Y} (h : e.IsImage s t) (h' : e.IsImage s' t') :
e.IsImage (s \ s') (t \ t')
theorem PartialHomeomorph.IsImage.leftInvOn_piecewise {X : Type u_1} {Y : Type u_3} [] [] {e : } {s : Set X} {t : Set Y} {e' : } [(i : X) → Decidable (i s)] [(i : Y) → Decidable (i t)] (h : e.IsImage s t) (h' : e'.IsImage s t) :
Set.LeftInvOn (t.piecewise e.symm e'.symm) (s.piecewise e e') (s.ite e.source e'.source)
theorem PartialHomeomorph.IsImage.inter_eq_of_inter_eq_of_eqOn {X : Type u_1} {Y : Type u_3} [] [] {e : } {s : Set X} {t : Set Y} {e' : } (h : e.IsImage s t) (h' : e'.IsImage 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 {X : Type u_1} {Y : Type u_3} [] [] {e : } {s : Set X} {t : Set Y} {e' : } (h : e.IsImage s t) (hs : e.source s = e'.source s) (Heq : Set.EqOn (↑e) (↑e') (e.source s)) :
Set.EqOn (↑e.symm) (↑e'.symm) (e.target t)
theorem PartialHomeomorph.IsImage.map_nhdsWithin_eq {X : Type u_1} {Y : Type u_3} [] [] {e : } {s : Set X} {t : Set Y} {x : X} (h : e.IsImage s t) (hx : x e.source) :
Filter.map (↑e) (nhdsWithin x s) = nhdsWithin (e x) t
theorem PartialHomeomorph.IsImage.closure {X : Type u_1} {Y : Type u_3} [] [] {e : } {s : Set X} {t : Set Y} (h : e.IsImage s t) :
e.IsImage (closure s) (closure t)
theorem PartialHomeomorph.IsImage.interior {X : Type u_1} {Y : Type u_3} [] [] {e : } {s : Set X} {t : Set Y} (h : e.IsImage s t) :
e.IsImage (interior s) (interior t)
theorem PartialHomeomorph.IsImage.frontier {X : Type u_1} {Y : Type u_3} [] [] {e : } {s : Set X} {t : Set Y} (h : e.IsImage s t) :
e.IsImage (frontier s) (frontier t)
theorem PartialHomeomorph.IsImage.isOpen_iff {X : Type u_1} {Y : Type u_3} [] [] {e : } {s : Set X} {t : Set Y} (h : e.IsImage s t) :
IsOpen (e.source s) IsOpen (e.target t)
@[simp]
theorem PartialHomeomorph.IsImage.restr_toPartialEquiv {X : Type u_1} {Y : Type u_3} [] [] {e : } {s : Set X} {t : Set Y} (h : e.IsImage s t) (hs : IsOpen (e.source s)) :
(h.restr hs).toPartialEquiv = .restr
def PartialHomeomorph.IsImage.restr {X : Type u_1} {Y : Type u_3} [] [] {e : } {s : Set X} {t : Set Y} (h : e.IsImage s t) (hs : IsOpen (e.source s)) :

Restrict a PartialHomeomorph to a pair of corresponding open sets.

Equations
• h.restr hs = { toPartialEquiv := .restr, open_source := hs, open_target := , continuousOn_toFun := , continuousOn_invFun := }
Instances For
theorem PartialHomeomorph.isImage_source_target {X : Type u_1} {Y : Type u_3} [] [] (e : ) :
e.IsImage e.source e.target
theorem PartialHomeomorph.isImage_source_target_of_disjoint {X : Type u_1} {Y : Type u_3} [] [] (e : ) (e' : ) (hs : Disjoint e.source e'.source) (ht : Disjoint e.target e'.target) :
e.IsImage e'.source e'.target
theorem PartialHomeomorph.preimage_interior {X : Type u_1} {Y : Type u_3} [] [] (e : ) (s : Set Y) :
e.source e ⁻¹' = 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 {X : Type u_1} {Y : Type u_3} [] [] (e : ) (s : Set Y) :
e.source e ⁻¹' = e.source closure (e ⁻¹' s)
theorem PartialHomeomorph.preimage_frontier {X : Type u_1} {Y : Type u_3} [] [] (e : ) (s : Set Y) :
e.source e ⁻¹' = e.source frontier (e ⁻¹' s)
def PartialHomeomorph.ofContinuousOpenRestrict {X : Type u_1} {Y : Type u_3} [] [] (e : ) (hc : ContinuousOn (↑e) e.source) (ho : IsOpenMap (e.source.restrict e)) (hs : IsOpen e.source) :

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

Equations
• = { toPartialEquiv := e, open_source := hs, open_target := , continuousOn_toFun := hc, continuousOn_invFun := }
Instances For
def PartialHomeomorph.ofContinuousOpen {X : Type u_1} {Y : Type u_3} [] [] (e : ) (hc : ContinuousOn (↑e) e.source) (ho : ) (hs : IsOpen e.source) :

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

Equations
Instances For
def PartialHomeomorph.restrOpen {X : Type u_1} {Y : Type u_3} [] [] (e : ) (s : Set X) (hs : ) :

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
• e.restrOpen s hs = .restr
Instances For
@[simp]
theorem PartialHomeomorph.restrOpen_toPartialEquiv {X : Type u_1} {Y : Type u_3} [] [] (e : ) (s : Set X) (hs : ) :
(e.restrOpen s hs).toPartialEquiv = e.restr s
theorem PartialHomeomorph.restrOpen_source {X : Type u_1} {Y : Type u_3} [] [] (e : ) (s : Set X) (hs : ) :
(e.restrOpen s hs).source = e.source s
@[simp]
theorem PartialHomeomorph.restr_symm_apply {X : Type u_1} {Y : Type u_3} [] [] (e : ) (s : Set X) :
(e.restr s).symm = e.symm
@[simp]
theorem PartialHomeomorph.restr_apply {X : Type u_1} {Y : Type u_3} [] [] (e : ) (s : Set X) :
(e.restr s) = e
theorem PartialHomeomorph.restr_target {X : Type u_1} {Y : Type u_3} [] [] (e : ) (s : Set X) :
(e.restr s).target = e.target e.symm ⁻¹'
theorem PartialHomeomorph.restr_source {X : Type u_1} {Y : Type u_3} [] [] (e : ) (s : Set X) :
(e.restr s).source = e.source
def PartialHomeomorph.restr {X : Type u_1} {Y : Type u_3} [] [] (e : ) (s : Set X) :

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
Instances For
@[simp]
theorem PartialHomeomorph.restr_toPartialEquiv {X : Type u_1} {Y : Type u_3} [] [] (e : ) (s : Set X) :
(e.restr s).toPartialEquiv = e.restr (interior s)
theorem PartialHomeomorph.restr_source' {X : Type u_1} {Y : Type u_3} [] [] (e : ) (s : Set X) (hs : ) :
(e.restr s).source = e.source s
theorem PartialHomeomorph.restr_toPartialEquiv' {X : Type u_1} {Y : Type u_3} [] [] (e : ) (s : Set X) (hs : ) :
(e.restr s).toPartialEquiv = e.restr s
theorem PartialHomeomorph.restr_eq_of_source_subset {X : Type u_1} {Y : Type u_3} [] [] {e : } {s : Set X} (h : e.source s) :
e.restr s = e
@[simp]
theorem PartialHomeomorph.restr_univ {X : Type u_1} {Y : Type u_3} [] [] {e : } :
e.restr Set.univ = e
theorem PartialHomeomorph.restr_source_inter {X : Type u_1} {Y : Type u_3} [] [] (e : ) (s : Set X) :
e.restr (e.source s) = e.restr s
@[simp]
theorem PartialHomeomorph.refl_apply (X : Type u_7) [] :
= id
theorem PartialHomeomorph.refl_target (X : Type u_7) [] :
.target = Set.univ
theorem PartialHomeomorph.refl_source (X : Type u_7) [] :
.source = Set.univ
def PartialHomeomorph.refl (X : Type u_7) [] :

The identity on the whole space as a partial homeomorphism.

Equations
• = .toPartialHomeomorph
Instances For
@[simp]
theorem PartialHomeomorph.refl_partialEquiv {X : Type u_1} [] :
.toPartialEquiv =
@[simp]
theorem PartialHomeomorph.refl_symm {X : Type u_1} [] :

ofSet: the identity on a set s

@[simp]
theorem PartialHomeomorph.ofSet_apply {X : Type u_1} [] (s : Set X) (hs : ) :
= id
theorem PartialHomeomorph.ofSet_target {X : Type u_1} [] (s : Set X) (hs : ) :
.target = s
theorem PartialHomeomorph.ofSet_source {X : Type u_1} [] (s : Set X) (hs : ) :
.source = s
def PartialHomeomorph.ofSet {X : Type u_1} [] (s : Set X) (hs : ) :

The identity partial equivalence on a set s

Equations
• = { toPartialEquiv := , open_source := hs, open_target := hs, continuousOn_toFun := , continuousOn_invFun := }
Instances For
@[simp]
theorem PartialHomeomorph.ofSet_toPartialEquiv {X : Type u_1} [] {s : Set X} (hs : ) :
.toPartialEquiv =
@[simp]
theorem PartialHomeomorph.ofSet_symm {X : Type u_1} [] {s : Set X} (hs : ) :
.symm =
@[simp]

trans: composition of two partial homeomorphisms

@[simp]
theorem PartialHomeomorph.trans'_symm_apply {X : Type u_1} {Y : Type u_3} {Z : Type u_5} [] [] [] (e : ) (e' : ) (h : e.target = e'.source) :
∀ (a : Z), (e.trans' e' h).symm a = e.symm (e'.symm a)
@[simp]
theorem PartialHomeomorph.trans'_toPartialEquiv {X : Type u_1} {Y : Type u_3} {Z : Type u_5} [] [] [] (e : ) (e' : ) (h : e.target = e'.source) :
(e.trans' e' h).toPartialEquiv = e.trans' e'.toPartialEquiv h
@[simp]
theorem PartialHomeomorph.trans'_apply {X : Type u_1} {Y : Type u_3} {Z : Type u_5} [] [] [] (e : ) (e' : ) (h : e.target = e'.source) :
∀ (a : X), (e.trans' e' h) a = e' (e a)
theorem PartialHomeomorph.trans'_source {X : Type u_1} {Y : Type u_3} {Z : Type u_5} [] [] [] (e : ) (e' : ) (h : e.target = e'.source) :
(e.trans' e' h).source = e.source
theorem PartialHomeomorph.trans'_target {X : Type u_1} {Y : Type u_3} {Z : Type u_5} [] [] [] (e : ) (e' : ) (h : e.target = e'.source) :
(e.trans' e' h).target = e'.target
def PartialHomeomorph.trans' {X : Type u_1} {Y : Type u_3} {Z : Type u_5} [] [] [] (e : ) (e' : ) (h : e.target = e'.source) :

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

Equations
• e.trans' e' h = { toPartialEquiv := e.trans' e'.toPartialEquiv h, open_source := , open_target := , continuousOn_toFun := , continuousOn_invFun := }
Instances For
def PartialHomeomorph.trans {X : Type u_1} {Y : Type u_3} {Z : Type u_5} [] [] [] (e : ) (e' : ) :

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

Equations
• e.trans e' = (e.symm.restrOpen e'.source ).symm.trans' (e'.restrOpen e.target )
Instances For
@[simp]
theorem PartialHomeomorph.trans_toPartialEquiv {X : Type u_1} {Y : Type u_3} {Z : Type u_5} [] [] [] (e : ) (e' : ) :
(e.trans e').toPartialEquiv = e.trans e'.toPartialEquiv
@[simp]
theorem PartialHomeomorph.coe_trans {X : Type u_1} {Y : Type u_3} {Z : Type u_5} [] [] [] (e : ) (e' : ) :
(e.trans e') = e' e
@[simp]
theorem PartialHomeomorph.coe_trans_symm {X : Type u_1} {Y : Type u_3} {Z : Type u_5} [] [] [] (e : ) (e' : ) :
(e.trans e').symm = e.symm e'.symm
theorem PartialHomeomorph.trans_apply {X : Type u_1} {Y : Type u_3} {Z : Type u_5} [] [] [] (e : ) (e' : ) {x : X} :
(e.trans e') x = e' (e x)
theorem PartialHomeomorph.trans_symm_eq_symm_trans_symm {X : Type u_1} {Y : Type u_3} {Z : Type u_5} [] [] [] (e : ) (e' : ) :
(e.trans e').symm = e'.symm.trans e.symm
theorem PartialHomeomorph.trans_source {X : Type u_1} {Y : Type u_3} {Z : Type u_5} [] [] [] (e : ) (e' : ) :
(e.trans e').source = e.source e ⁻¹' e'.source
theorem PartialHomeomorph.trans_source' {X : Type u_1} {Y : Type u_3} {Z : Type u_5} [] [] [] (e : ) (e' : ) :
(e.trans e').source = e.source e ⁻¹' (e.target e'.source)
theorem PartialHomeomorph.trans_source'' {X : Type u_1} {Y : Type u_3} {Z : Type u_5} [] [] [] (e : ) (e' : ) :
(e.trans e').source = e.symm '' (e.target e'.source)
theorem PartialHomeomorph.image_trans_source {X : Type u_1} {Y : Type u_3} {Z : Type u_5} [] [] [] (e : ) (e' : ) :
e '' (e.trans e').source = e.target e'.source
theorem PartialHomeomorph.trans_target {X : Type u_1} {Y : Type u_3} {Z : Type u_5} [] [] [] (e : ) (e' : ) :
(e.trans e').target = e'.target e'.symm ⁻¹' e.target
theorem PartialHomeomorph.trans_target' {X : Type u_1} {Y : Type u_3} {Z : Type u_5} [] [] [] (e : ) (e' : ) :
(e.trans e').target = e'.target e'.symm ⁻¹' (e'.source e.target)
theorem PartialHomeomorph.trans_target'' {X : Type u_1} {Y : Type u_3} {Z : Type u_5} [] [] [] (e : ) (e' : ) :
(e.trans e').target = e' '' (e'.source e.target)
theorem PartialHomeomorph.inv_image_trans_target {X : Type u_1} {Y : Type u_3} {Z : Type u_5} [] [] [] (e : ) (e' : ) :
e'.symm '' (e.trans e').target = e'.source e.target
theorem PartialHomeomorph.trans_assoc {X : Type u_1} {Y : Type u_3} {Z : Type u_5} {Z' : Type u_6} [] [] [] [] (e : ) (e' : ) (e'' : ) :
(e.trans e').trans e'' = e.trans (e'.trans e'')
@[simp]
theorem PartialHomeomorph.trans_refl {X : Type u_1} {Y : Type u_3} [] [] (e : ) :
e.trans = e
@[simp]
theorem PartialHomeomorph.refl_trans {X : Type u_1} {Y : Type u_3} [] [] (e : ) :
.trans e = e
theorem PartialHomeomorph.trans_ofSet {X : Type u_1} {Y : Type u_3} [] [] (e : ) {s : Set Y} (hs : ) :
e.trans = e.restr (e ⁻¹' s)
theorem PartialHomeomorph.trans_of_set' {X : Type u_1} {Y : Type u_3} [] [] (e : ) {s : Set Y} (hs : ) :
e.trans = e.restr (e.source e ⁻¹' s)
theorem PartialHomeomorph.ofSet_trans {X : Type u_1} {Y : Type u_3} [] [] (e : ) {s : Set X} (hs : ) :
.trans e = e.restr s
theorem PartialHomeomorph.ofSet_trans' {X : Type u_1} {Y : Type u_3} [] [] (e : ) {s : Set X} (hs : ) :
.trans e = e.restr (e.source s)
@[simp]
theorem PartialHomeomorph.ofSet_trans_ofSet {X : Type u_1} [] {s : Set X} (hs : ) {s' : Set X} (hs' : IsOpen s') :
theorem PartialHomeomorph.restr_trans {X : Type u_1} {Y : Type u_3} {Z : Type u_5} [] [] [] (e : ) (e' : ) (s : Set X) :
(e.restr s).trans e' = (e.trans e').restr s

EqOnSource: equivalence on their source

def PartialHomeomorph.EqOnSource {X : Type u_1} {Y : Type u_3} [] [] (e : ) (e' : ) :

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
• e.EqOnSource e' = (e.source = e'.source Set.EqOn (↑e) (↑e') e.source)
Instances For
theorem PartialHomeomorph.eqOnSource_iff {X : Type u_1} {Y : Type u_3} [] [] (e : ) (e' : ) :
e.EqOnSource e' e.EqOnSource e'.toPartialEquiv
instance PartialHomeomorph.eqOnSourceSetoid {X : Type u_1} {Y : Type u_3} [] [] :

EqOnSource is an equivalence relation.

Equations
• PartialHomeomorph.eqOnSourceSetoid = { r := PartialHomeomorph.EqOnSource, iseqv := }
theorem PartialHomeomorph.eqOnSource_refl {X : Type u_1} {Y : Type u_3} [] [] (e : ) :
e e
theorem PartialHomeomorph.EqOnSource.symm' {X : Type u_1} {Y : Type u_3} [] [] {e : } {e' : } (h : e e') :
e.symm e'.symm

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

theorem PartialHomeomorph.EqOnSource.source_eq {X : Type u_1} {Y : Type u_3} [] [] {e : } {e' : } (h : e e') :
e.source = e'.source

Two equivalent partial homeomorphisms have the same source.

theorem PartialHomeomorph.EqOnSource.target_eq {X : Type u_1} {Y : Type u_3} [] [] {e : } {e' : } (h : e e') :
e.target = e'.target

Two equivalent partial homeomorphisms have the same target.

theorem PartialHomeomorph.EqOnSource.eqOn {X : Type u_1} {Y : Type u_3} [] [] {e : } {e' : } (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 {X : Type u_1} {Y : Type u_3} [] [] {e : } {e' : } (h : e e') :
Set.EqOn (↑e.symm) (↑e'.symm) e.target

Two equivalent partial homeomorphisms have coinciding invFun on the target

theorem PartialHomeomorph.EqOnSource.trans' {X : Type u_1} {Y : Type u_3} {Z : Type u_5} [] [] [] {e : } {e' : } {f : } {f' : } (he : e e') (hf : f f') :
e.trans f e'.trans f'

Composition of partial homeomorphisms respects equivalence.

theorem PartialHomeomorph.EqOnSource.restr {X : Type u_1} {Y : Type u_3} [] [] {e : } {e' : } (he : e e') (s : Set X) :
e.restr s e'.restr s

Restriction of partial homeomorphisms respects equivalence

theorem PartialHomeomorph.Set.EqOn.restr_eqOn_source {X : Type u_1} {Y : Type u_3} [] [] {e : } {e' : } (h : Set.EqOn (↑e) (↑e') (e.source e'.source)) :
e.restr e'.source e'.restr e.source

Two equivalent partial homeomorphisms are equal when the source and target are univ.

theorem PartialHomeomorph.self_trans_symm {X : Type u_1} {Y : Type u_3} [] [] (e : ) :
e.trans e.symm PartialHomeomorph.ofSet 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 {X : Type u_1} {Y : Type u_3} [] [] (e : ) :
e.symm.trans e PartialHomeomorph.ofSet e.target
theorem PartialHomeomorph.eq_of_eqOnSource_univ {X : Type u_1} {Y : Type u_3} [] [] {e : } {e' : } (h : e e') (s : e.source = Set.univ) (t : e.target = Set.univ) :
e = e'

product of two partial homeomorphisms

@[simp]
theorem PartialHomeomorph.prod_toPartialEquiv {X : Type u_1} {X' : Type u_2} {Y : Type u_3} {Y' : Type u_4} [] [] [] [] (eX : ) (eY : ) :
(eX.prod eY).toPartialEquiv = eX.prod eY.toPartialEquiv
@[simp]
theorem PartialHomeomorph.prod_apply {X : Type u_1} {X' : Type u_2} {Y : Type u_3} {Y' : Type u_4} [] [] [] [] (eX : ) (eY : ) :
(eX.prod eY) = fun (p : X × Y) => (eX p.1, eY p.2)
theorem PartialHomeomorph.prod_symm_apply {X : Type u_1} {X' : Type u_2} {Y : Type u_3} {Y' : Type u_4} [] [] [] [] (eX : ) (eY : ) (p : X' × Y') :
(eX.prod eY).symm p = (eX.symm p.1, eY.symm p.2)
theorem PartialHomeomorph.prod_target {X : Type u_1} {X' : Type u_2} {Y : Type u_3} {Y' : Type u_4} [] [] [] [] (eX : ) (eY : ) :
(eX.prod eY).target = eX.target ×ˢ eY.target
theorem PartialHomeomorph.prod_source {X : Type u_1} {X' : Type u_2} {Y : Type u_3} {Y' : Type u_4} [] [] [] [] (eX : ) (eY : ) :
(eX.prod eY).source = eX.source ×ˢ eY.source
def PartialHomeomorph.prod {X : Type u_1} {X' : Type u_2} {Y : Type u_3} {Y' : Type u_4} [] [] [] [] (eX : ) (eY : ) :
PartialHomeomorph (X × Y) (X' × Y')

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

Equations
• eX.prod eY = { toPartialEquiv := eX.prod eY.toPartialEquiv, open_source := , open_target := , continuousOn_toFun := , continuousOn_invFun := }
Instances For
@[simp]
theorem PartialHomeomorph.prod_symm {X : Type u_1} {X' : Type u_2} {Y : Type u_3} {Y' : Type u_4} [] [] [] [] (eX : ) (eY : ) :
(eX.prod eY).symm = eX.symm.prod eY.symm
@[simp]
theorem PartialHomeomorph.refl_prod_refl {X : Type u_1} {Y : Type u_3} [] [] :
@[simp]
theorem PartialHomeomorph.prod_trans {X : Type u_1} {X' : Type u_2} {Y : Type u_3} {Y' : Type u_4} {Z : Type u_5} {Z' : Type u_6} [] [] [] [] [] [] (e : ) (f : ) (e' : ) (f' : ) :
(e.prod e').trans (f.prod f') = (e.trans f).prod (e'.trans f')
theorem PartialHomeomorph.prod_eq_prod_of_nonempty {X : Type u_1} {X' : Type u_2} {Y : Type u_3} {Y' : Type u_4} [] [] [] [] {eX : } {eX' : } {eY : } {eY' : } (h : (eX.prod eY).source.Nonempty) :
eX.prod eY = eX'.prod eY' eX = eX' eY = eY'
theorem PartialHomeomorph.prod_eq_prod_of_nonempty' {X : Type u_1} {X' : Type u_2} {Y : Type u_3} {Y' : Type u_4} [] [] [] [] {eX : } {eX' : } {eY : } {eY' : } (h : (eX'.prod eY').source.Nonempty) :
eX.prod eY = eX'.prod eY' eX = eX' eY = eY'

finite product of partial homeomorphisms

@[simp]
theorem PartialHomeomorph.pi_toPartialEquiv {ι : Type u_7} [] {X : ιType u_8} {Y : ιType u_9} [(i : ι) → TopologicalSpace (X i)] [(i : ι) → TopologicalSpace (Y i)] (ei : (i : ι) → PartialHomeomorph (X i) (Y i)) :
.toPartialEquiv = PartialEquiv.pi fun (i : ι) => (ei i).toPartialEquiv
def PartialHomeomorph.pi {ι : Type u_7} [] {X : ιType u_8} {Y : ιType u_9} [(i : ι) → TopologicalSpace (X i)] [(i : ι) → TopologicalSpace (Y i)] (ei : (i : ι) → PartialHomeomorph (X i) (Y i)) :
PartialHomeomorph ((i : ι) → X i) ((i : ι) → Y i)

The product of a finite family of PartialHomeomorphs.

Equations
• = { toPartialEquiv := PartialEquiv.pi fun (i : ι) => (ei i).toPartialEquiv, open_source := , open_target := , continuousOn_toFun := , continuousOn_invFun := }
Instances For

combining two partial homeomorphisms using Set.piecewise

@[simp]
theorem PartialHomeomorph.piecewise_apply {X : Type u_1} {Y : Type u_3} [] [] (e : ) (e' : ) (s : Set X) (t : Set Y) [(x : X) → Decidable (x s)] [(y : Y) → Decidable (y t)] (H : e.IsImage s t) (H' : e'.IsImage s t) (Hs : e.source = e'.source ) (Heq : Set.EqOn (↑e) (↑e') (e.source )) :
(e.piecewise e' s t H H' Hs Heq) = s.piecewise e e'
@[simp]
theorem PartialHomeomorph.piecewise_toPartialEquiv {X : Type u_1} {Y : Type u_3} [] [] (e : ) (e' : ) (s : Set X) (t : Set Y) [(x : X) → Decidable (x s)] [(y : Y) → Decidable (y t)] (H : e.IsImage s t) (H' : e'.IsImage s t) (Hs : e.source = e'.source ) (Heq : Set.EqOn (↑e) (↑e') (e.source )) :
(e.piecewise e' s t H H' Hs Heq).toPartialEquiv = e.piecewise e'.toPartialEquiv s t H H'
def PartialHomeomorph.piecewise {X : Type u_1} {Y : Type u_3} [] [] (e : ) (e' : ) (s : Set X) (t : Set Y) [(x : X) → Decidable (x s)] [(y : Y) → Decidable (y t)] (H : e.IsImage s t) (H' : e'.IsImage s t) (Hs : e.source = e'.source ) (Heq : Set.EqOn (↑e) (↑e') (e.source )) :

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
• e.piecewise e' s t H H' Hs Heq = { toPartialEquiv := e.piecewise e'.toPartialEquiv s t H H', open_source := , open_target := , continuousOn_toFun := , continuousOn_invFun := }
Instances For
@[simp]
theorem PartialHomeomorph.symm_piecewise {X : Type u_1} {Y : Type u_3} [] [] (e : ) (e' : ) {s : Set X} {t : Set Y} [(x : X) → Decidable (x s)] [(y : Y) → Decidable (y t)] (H : e.IsImage s t) (H' : e'.IsImage s t) (Hs : e.source = e'.source ) (Heq : Set.EqOn (↑e) (↑e') (e.source )) :
(e.piecewise e' s t H H' Hs Heq).symm = e.symm.piecewise e'.symm t s
def PartialHomeomorph.disjointUnion {X : Type u_1} {Y : Type u_3} [] [] (e : ) (e' : ) [(x : X) → Decidable (x e.source)] [(y : Y) → Decidable (y e.target)] (Hs : Disjoint e.source e'.source) (Ht : Disjoint e.target e'.target) :

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.

Equations
• e.disjointUnion e' Hs Ht = (e.piecewise e' e.source e.target ).replaceEquiv (e.disjointUnion e'.toPartialEquiv Hs Ht)
Instances For
theorem PartialHomeomorph.continuousWithinAt_iff_continuousWithinAt_comp_right {X : Type u_1} {Y : Type u_3} {Z : Type u_5} [] [] [] (e : ) {f : YZ} {s : Set Y} {x : Y} (h : x e.target) :
ContinuousWithinAt (f e) (e ⁻¹' s) (e.symm 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 {X : Type u_1} {Y : Type u_3} {Z : Type u_5} [] [] [] (e : ) {f : YZ} {x : Y} (h : x e.target) :
ContinuousAt (f e) (e.symm 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 {X : Type u_1} {Y : Type u_3} {Z : Type u_5} [] [] [] (e : ) {f : YZ} {s : Set Y} (h : s e.target) :
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 {X : Type u_1} {Y : Type u_3} {Z : Type u_5} [] [] [] (e : ) {f : ZX} {s : Set Z} {x : Z} (hx : f x e.source) (h : f ⁻¹' e.source ) :
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 {X : Type u_1} {Y : Type u_3} {Z : Type u_5} [] [] [] (e : ) {f : ZX} {x : Z} (h : f ⁻¹' e.source nhds 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 {X : Type u_1} {Y : Type u_3} {Z : Type u_5} [] [] [] (e : ) {f : ZX} {s : Set Z} (h : s f ⁻¹' e.source) :
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 {X : Type u_1} {Y : Type u_3} {Z : Type u_5} [] [] [] (e : ) {f : ZX} (h : f ⁻¹' e.source = Set.univ) :
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_symm_apply {X : Type u_1} {Y : Type u_3} [] [] (e : ) {s : Set X} {t : Set Y} (hs : s e.source) (ht : e '' s = t) :
∀ (a : t), (e.homeomorphOfImageSubsetSource hs ht).symm a = Set.MapsTo.restrict (↑e.symm) t s a
@[simp]
theorem PartialHomeomorph.homeomorphOfImageSubsetSource_apply {X : Type u_1} {Y : Type u_3} [] [] (e : ) {s : Set X} {t : Set Y} (hs : s e.source) (ht : e '' s = t) :
∀ (a : s), (e.homeomorphOfImageSubsetSource hs ht) a = Set.MapsTo.restrict (↑e) s t a
def PartialHomeomorph.homeomorphOfImageSubsetSource {X : Type u_1} {Y : Type u_3} [] [] (e : ) {s : Set X} {t : Set Y} (hs : s e.source) (ht : e '' s = t) :
s ≃ₜ t

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

Equations
• One or more equations did not get rendered due to their size.
Instances For
@[simp]
theorem PartialHomeomorph.toHomeomorphSourceTarget_apply_coe {X : Type u_1} {Y : Type u_3} [] [] (e : ) :
∀ (a : e.source), (e.toHomeomorphSourceTarget a) = e a
@[simp]
theorem PartialHomeomorph.toHomeomorphSourceTarget_symm_apply_coe {X : Type u_1} {Y : Type u_3} [] [] (e : ) :
∀ (a : e.target), (e.toHomeomorphSourceTarget.symm a) = e.symm a
def PartialHomeomorph.toHomeomorphSourceTarget {X : Type u_1} {Y : Type u_3} [] [] (e : ) :
e.source ≃ₜ e.target

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

Equations
• e.toHomeomorphSourceTarget = e.homeomorphOfImageSubsetSource
Instances For
theorem PartialHomeomorph.secondCountableTopology_source {X : Type u_1} {Y : Type u_3} [] [] (e : ) :
theorem PartialHomeomorph.nhds_eq_comap_inf_principal {X : Type u_1} {Y : Type u_3} [] [] (e : ) {x : X} (hx : x e.source) :
nhds x = Filter.comap (↑e) (nhds (e x)) Filter.principal e.source
@[simp]
theorem PartialHomeomorph.toHomeomorphOfSourceEqUnivTargetEqUniv_symm_apply {X : Type u_1} {Y : Type u_3} [] [] (e : ) (h : e.source = Set.univ) (h' : e.target = Set.univ) :
(e.toHomeomorphOfSourceEqUnivTargetEqUniv h h').symm = e.symm
@[simp]
theorem PartialHomeomorph.toHomeomorphOfSourceEqUnivTargetEqUniv_apply {X : Type u_1} {Y : Type u_3} [] [] (e : ) (h : e.source = Set.univ) (h' : e.target = Set.univ) :
(e.toHomeomorphOfSourceEqUnivTargetEqUniv h h') = e
def PartialHomeomorph.toHomeomorphOfSourceEqUnivTargetEqUniv {X : Type u_1} {Y : Type u_3} [] [] (e : ) (h : e.source = Set.univ) (h' : e.target = Set.univ) :
X ≃ₜ Y

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

Equations
• e.toHomeomorphOfSourceEqUnivTargetEqUniv h h' = { toFun := e, invFun := e.symm, left_inv := , right_inv := , continuous_toFun := , continuous_invFun := }
Instances For
theorem PartialHomeomorph.openEmbedding_restrict {X : Type u_1} {Y : Type u_3} [] [] (e : ) :
OpenEmbedding (e.source.restrict e)
theorem PartialHomeomorph.to_openEmbedding {X : Type u_1} {Y : Type u_3} [] [] (e : ) (h : e.source = Set.univ) :

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

@[simp]
theorem Homeomorph.refl_toPartialHomeomorph {X : Type u_1} [] :
.toPartialHomeomorph =
@[simp]
theorem Homeomorph.symm_toPartialHomeomorph {X : Type u_1} {Y : Type u_3} [] [] (e : X ≃ₜ Y) :
e.symm.toPartialHomeomorph = e.toPartialHomeomorph.symm
@[simp]
theorem Homeomorph.trans_toPartialHomeomorph {X : Type u_1} {Y : Type u_3} {Z : Type u_5} [] [] [] (e : X ≃ₜ Y) (e' : Y ≃ₜ Z) :
(e.trans e').toPartialHomeomorph = e.toPartialHomeomorph.trans e'.toPartialHomeomorph
@[simp]
theorem Homeomorph.transPartialHomeomorph_source {X : Type u_1} {Y : Type u_3} {Z : Type u_5} [] [] [] (e : X ≃ₜ Y) (f' : ) :
(e.transPartialHomeomorph f').source = e ⁻¹' f'.source
@[simp]
theorem Homeomorph.transPartialHomeomorph_target {X : Type u_1} {Y : Type u_3} {Z : Type u_5} [] [] [] (e : X ≃ₜ Y) (f' : ) :
(e.transPartialHomeomorph f').target = f'.target
@[simp]
theorem Homeomorph.transPartialHomeomorph_apply {X : Type u_1} {Y : Type u_3} {Z : Type u_5} [] [] [] (e : X ≃ₜ Y) (f' : ) :
(e.transPartialHomeomorph f') = f' e
@[simp]
theorem Homeomorph.transPartialHomeomorph_symm_apply {X : Type u_1} {Y : Type u_3} {Z : Type u_5} [] [] [] (e : X ≃ₜ Y) (f' : ) :
(e.transPartialHomeomorph f').symm = e.symm f'.symm
def Homeomorph.transPartialHomeomorph {X : Type u_1} {Y : Type u_3} {Z : Type u_5} [] [] [] (e : X ≃ₜ Y) (f' : ) :

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

Equations
• e.transPartialHomeomorph f' = { toPartialEquiv := e.transPartialEquiv f'.toPartialEquiv, open_source := , open_target := , continuousOn_toFun := , continuousOn_invFun := }
Instances For
theorem Homeomorph.transPartialHomeomorph_eq_trans {X : Type u_1} {Y : Type u_3} {Z : Type u_5} [] [] [] (e : X ≃ₜ Y) (f' : ) :
e.transPartialHomeomorph f' = e.toPartialHomeomorph.trans f'
@[simp]
theorem Homeomorph.transPartialHomeomorph_trans {X : Type u_1} {Y : Type u_3} {Z : Type u_5} {Z' : Type u_6} [] [] [] [] (e : X ≃ₜ Y) (f : ) (f' : ) :
(e.transPartialHomeomorph f).trans f' = e.transPartialHomeomorph (f.trans f')
@[simp]
theorem Homeomorph.trans_transPartialHomeomorph {X : Type u_1} {Y : Type u_3} {Z : Type u_5} {Z' : Type u_6} [] [] [] [] (e : X ≃ₜ Y) (e' : Y ≃ₜ Z) (f'' : ) :
(e.trans e').transPartialHomeomorph f'' = e.transPartialHomeomorph (e'.transPartialHomeomorph f'')
@[simp]
theorem OpenEmbedding.toPartialHomeomorph_apply {X : Type u_1} {Y : Type u_3} [] [] (f : XY) (h : ) [] :
@[simp]
theorem OpenEmbedding.toPartialHomeomorph_target {X : Type u_1} {Y : Type u_3} [] [] (f : XY) (h : ) [] :
.target =
@[simp]
theorem OpenEmbedding.toPartialHomeomorph_source {X : Type u_1} {Y : Type u_3} [] [] (f : XY) (h : ) [] :
.source = Set.univ
noncomputable def OpenEmbedding.toPartialHomeomorph {X : Type u_1} {Y : Type u_3} [] [] (f : XY) (h : ) [] :

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

Equations
Instances For
theorem OpenEmbedding.toPartialHomeomorph_left_inv {X : Type u_1} {Y : Type u_3} [] [] (f : XY) (h : ) [] {x : X} :
.symm (f x) = x
theorem OpenEmbedding.toPartialHomeomorph_right_inv {X : Type u_1} {Y : Type u_3} [] [] (f : XY) (h : ) [] {x : Y} (hx : x ) :
f (.symm x) = x

inclusion of an open set in a topological space

noncomputable def TopologicalSpace.Opens.partialHomeomorphSubtypeCoe {X : Type u_1} [] (s : ) (hs : Nonempty { x : X // x s }) :
PartialHomeomorph { x : X // x s } X

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

Equations
Instances For
@[simp]
theorem TopologicalSpace.Opens.partialHomeomorphSubtypeCoe_coe {X : Type u_1} [] (s : ) (hs : Nonempty { x : X // x s }) :
(s.partialHomeomorphSubtypeCoe hs) = Subtype.val
@[simp]
theorem TopologicalSpace.Opens.partialHomeomorphSubtypeCoe_source {X : Type u_1} [] (s : ) (hs : Nonempty { x : X // x s }) :
(s.partialHomeomorphSubtypeCoe hs).source = Set.univ
@[simp]
theorem TopologicalSpace.Opens.partialHomeomorphSubtypeCoe_target {X : Type u_1} [] (s : ) (hs : Nonempty { x : X // x s }) :
(s.partialHomeomorphSubtypeCoe hs).target = s
@[simp]
theorem PartialHomeomorph.transHomeomorph_source {X : Type u_1} {Y : Type u_3} {Z : Type u_5} [] [] [] (e : ) (f' : Y ≃ₜ Z) :
(e.transHomeomorph f').source = e.source
@[simp]
theorem PartialHomeomorph.transHomeomorph_symm_apply {X : Type u_1} {Y : Type u_3} {Z : Type u_5} [] [] [] (e : ) (f' : Y ≃ₜ Z) :
(e.transHomeomorph f').symm = e.symm f'.symm
@[simp]
theorem PartialHomeomorph.transHomeomorph_apply {X : Type u_1} {Y : Type u_3} {Z : Type u_5} [] [] [] (e : ) (f' : Y ≃ₜ Z) :
(e.transHomeomorph f') = f' e
@[simp]
theorem PartialHomeomorph.transHomeomorph_target {X : Type u_1} {Y : Type u_3} {Z : Type u_5} [] [] [] (e : ) (f' : Y ≃ₜ Z) :
(e.transHomeomorph f').target = f'.symm ⁻¹' e.target
def PartialHomeomorph.transHomeomorph {X : Type u_1} {Y : Type u_3} {Z : Type u_5} [] [] [] (e : ) (f' : Y ≃ₜ Z) :

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

Equations
• e.transHomeomorph f' = { toPartialEquiv := e.transEquiv f'.toEquiv, open_source := , open_target := , continuousOn_toFun := , continuousOn_invFun := }
Instances For
theorem PartialHomeomorph.transHomeomorph_eq_trans {X : Type u_1} {Y : Type u_3} {Z : Type u_5} [] [] [] (e : ) (f' : Y ≃ₜ Z) :
e.transHomeomorph f' = e.trans f'.toPartialHomeomorph
@[simp]
theorem PartialHomeomorph.transHomeomorph_transHomeomorph {X : Type u_1} {Y : Type u_3} {Z : Type u_5} {Z' : Type u_6} [] [] [] [] (e : ) (f' : Y ≃ₜ Z) (f'' : Z ≃ₜ Z') :
(e.transHomeomorph f').transHomeomorph f'' = e.transHomeomorph (f'.trans f'')
@[simp]
theorem PartialHomeomorph.trans_transHomeomorph {X : Type u_1} {Y : Type u_3} {Z : Type u_5} {Z' : Type u_6} [] [] [] [] (e : ) (e' : ) (f'' : Z ≃ₜ Z') :
(e.trans e').transHomeomorph f'' = e.trans (e'.transHomeomorph f'')

subtypeRestr: restriction to a subtype

noncomputable def PartialHomeomorph.subtypeRestr {X : Type u_1} {Y : Type u_3} [] [] (e : ) {s : } (hs : Nonempty { x : X // x s }) :
PartialHomeomorph { x : X // x s } Y

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
• e.subtypeRestr hs = (s.partialHomeomorphSubtypeCoe hs).trans e
Instances For
theorem PartialHomeomorph.subtypeRestr_def {X : Type u_1} {Y : Type u_3} [] [] (e : ) {s : } (hs : Nonempty { x : X // x s }) :
e.subtypeRestr hs = (s.partialHomeomorphSubtypeCoe hs).trans e
@[simp]
theorem PartialHomeomorph.subtypeRestr_coe {X : Type u_1} {Y : Type u_3} [] [] (e : ) {s : } (hs : Nonempty { x : X // x s }) :
(e.subtypeRestr hs) = (↑s).restrict e
@[simp]
theorem PartialHomeomorph.subtypeRestr_source {X : Type u_1} {Y : Type u_3} [] [] (e : ) {s : } (hs : Nonempty { x : X // x s }) :
(e.subtypeRestr hs).source = Subtype.val ⁻¹' e.source
theorem PartialHomeomorph.map_subtype_source {X : Type u_1} {Y : Type u_3} [] [] (e : ) {s : } (hs : Nonempty { x : X // x s }) {x : { x : X // x s }} (hxe : x e.source) :
e x (e.subtypeRestr hs).target
theorem PartialHomeomorph.subtypeRestr_symm_trans_subtypeRestr {X : Type u_1} {Y : Type u_3} [] [] {s : } (hs : Nonempty { x : X // x s }) (f : ) (f' : ) :
(f.subtypeRestr hs).symm.trans (f'.subtypeRestr hs) (f.symm.trans f').restr (f.target f.symm ⁻¹' s)

This lemma characterizes the transition functions of an open subset in terms of the transition functions of the original space.

theorem PartialHomeomorph.subtypeRestr_symm_eqOn {X : Type u_1} {Y : Type u_3} [] [] (e : ) {U : } (hU : Nonempty { x : X // x U }) :
Set.EqOn (↑e.symm) (Subtype.val (e.subtypeRestr hU).symm) (e.subtypeRestr hU).target
theorem PartialHomeomorph.subtypeRestr_symm_eqOn_of_le {X : Type u_1} {Y : Type u_3} [] [] (e : ) {U : } {V : } (hU : Nonempty { x : X // x U }) (hV : Nonempty { x : X // x V }) (hUV : U V) :
Set.EqOn (↑(e.subtypeRestr hV).symm) ( (e.subtypeRestr hU).symm) (e.subtypeRestr hU).target