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lift_openEmbedding_symm_source (e : OpenPartialHomeomorph X Z) (hf : IsOpenEmbedding f) :
(e.lift_openEmbedding hf).symm.source = e.target | rfl | lemma | OpenPartialHomeomorph.lift_openEmbedding_symm_source | Topology.OpenPartialHomeomorph | Mathlib/Topology/OpenPartialHomeomorph/Constructions.lean | [] | [
"OpenPartialHomeomorph"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
lift_openEmbedding_symm_target (e : OpenPartialHomeomorph X Z) (hf : IsOpenEmbedding f) :
(e.lift_openEmbedding hf).symm.target = f '' e.source | by
rw [OpenPartialHomeomorph.symm_target, e.lift_openEmbedding_source] | lemma | OpenPartialHomeomorph.lift_openEmbedding_symm_target | Topology.OpenPartialHomeomorph | Mathlib/Topology/OpenPartialHomeomorph/Constructions.lean | [] | [
"OpenPartialHomeomorph",
"OpenPartialHomeomorph.symm_target"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
lift_openEmbedding_trans_apply
(e e' : OpenPartialHomeomorph X Z) (hf : IsOpenEmbedding f) (z : Z) :
(e.lift_openEmbedding hf).symm.trans (e'.lift_openEmbedding hf) z = (e.symm.trans e') z | by
simp [hf.injective.extend_apply e'] | lemma | OpenPartialHomeomorph.lift_openEmbedding_trans_apply | Topology.OpenPartialHomeomorph | Mathlib/Topology/OpenPartialHomeomorph/Constructions.lean | [] | [
"OpenPartialHomeomorph"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
lift_openEmbedding_trans (e e' : OpenPartialHomeomorph X Z) (hf : IsOpenEmbedding f) :
(e.lift_openEmbedding hf).symm.trans (e'.lift_openEmbedding hf) = e.symm.trans e' | by
ext z
· exact e.lift_openEmbedding_trans_apply e' hf z
· simp [hf.injective.extend_apply e]
· simp_rw [OpenPartialHomeomorph.trans_source, e.lift_openEmbedding_symm_source, e.symm_source,
e.lift_openEmbedding_symm, e'.lift_openEmbedding_source]
refine ⟨fun ⟨hx, ⟨y, hy, hxy⟩⟩ ↦ ⟨hx, ?_⟩, fun ⟨hx, hx... | lemma | OpenPartialHomeomorph.lift_openEmbedding_trans | Topology.OpenPartialHomeomorph | Mathlib/Topology/OpenPartialHomeomorph/Constructions.lean | [] | [
"OpenPartialHomeomorph",
"OpenPartialHomeomorph.trans_source"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
eventually_left_inverse {x} (hx : x ∈ e.source) :
∀ᶠ y in 𝓝 x, e.symm (e y) = y | (e.open_source.eventually_mem hx).mono e.left_inv' | theorem | OpenPartialHomeomorph.eventually_left_inverse | Topology.OpenPartialHomeomorph | Mathlib/Topology/OpenPartialHomeomorph/Continuity.lean | [] | [] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
eventually_left_inverse' {x} (hx : x ∈ e.target) :
∀ᶠ y in 𝓝 (e.symm x), e.symm (e y) = y | e.eventually_left_inverse (e.map_target hx) | theorem | OpenPartialHomeomorph.eventually_left_inverse' | Topology.OpenPartialHomeomorph | Mathlib/Topology/OpenPartialHomeomorph/Continuity.lean | [] | [] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
eventually_right_inverse {x} (hx : x ∈ e.target) :
∀ᶠ y in 𝓝 x, e (e.symm y) = y | (e.open_target.eventually_mem hx).mono e.right_inv' | theorem | OpenPartialHomeomorph.eventually_right_inverse | Topology.OpenPartialHomeomorph | Mathlib/Topology/OpenPartialHomeomorph/Continuity.lean | [] | [] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
eventually_right_inverse' {x} (hx : x ∈ e.source) :
∀ᶠ y in 𝓝 (e x), e (e.symm y) = y | e.eventually_right_inverse (e.map_source hx) | theorem | OpenPartialHomeomorph.eventually_right_inverse' | Topology.OpenPartialHomeomorph | Mathlib/Topology/OpenPartialHomeomorph/Continuity.lean | [] | [] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
eventually_ne_nhdsWithin {x} (hx : x ∈ e.source) :
∀ᶠ x' in 𝓝[≠] x, e x' ≠ e x | eventually_nhdsWithin_iff.2 <|
(e.eventually_left_inverse hx).mono fun x' hx' =>
mt fun h => by rw [mem_singleton_iff, ← e.left_inv hx, ← h, hx'] | theorem | OpenPartialHomeomorph.eventually_ne_nhdsWithin | Topology.OpenPartialHomeomorph | Mathlib/Topology/OpenPartialHomeomorph/Continuity.lean | [] | [
"eventually_ne_nhdsWithin"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
nhdsWithin_source_inter {x} (hx : x ∈ e.source) (s : Set X) : 𝓝[e.source ∩ s] x = 𝓝[s] x | nhdsWithin_inter_of_mem (mem_nhdsWithin_of_mem_nhds <| IsOpen.mem_nhds e.open_source hx) | theorem | OpenPartialHomeomorph.nhdsWithin_source_inter | Topology.OpenPartialHomeomorph | Mathlib/Topology/OpenPartialHomeomorph/Continuity.lean | [] | [
"IsOpen.mem_nhds",
"Set",
"mem_nhdsWithin_of_mem_nhds",
"nhdsWithin_inter_of_mem"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
nhdsWithin_target_inter {x} (hx : x ∈ e.target) (s : Set Y) : 𝓝[e.target ∩ s] x = 𝓝[s] x | e.symm.nhdsWithin_source_inter hx s | theorem | OpenPartialHomeomorph.nhdsWithin_target_inter | Topology.OpenPartialHomeomorph | Mathlib/Topology/OpenPartialHomeomorph/Continuity.lean | [] | [
"Set"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
continuousAt {x : X} (h : x ∈ e.source) : ContinuousAt e x | (e.continuousOn x h).continuousAt (e.open_source.mem_nhds h) | theorem | OpenPartialHomeomorph.continuousAt | Topology.OpenPartialHomeomorph | Mathlib/Topology/OpenPartialHomeomorph/Continuity.lean | [] | [
"ContinuousAt"
] | An open partial homeomorphism is continuous at any point of its source | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
continuousAt_symm {x : Y} (h : x ∈ e.target) : ContinuousAt e.symm x | e.symm.continuousAt h | theorem | OpenPartialHomeomorph.continuousAt_symm | Topology.OpenPartialHomeomorph | Mathlib/Topology/OpenPartialHomeomorph/Continuity.lean | [] | [
"ContinuousAt"
] | An open partial homeomorphism inverse is continuous at any point of its target | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
tendsto_symm {x} (hx : x ∈ e.source) : Tendsto e.symm (𝓝 (e x)) (𝓝 x) | by
simpa only [ContinuousAt, e.left_inv hx] using e.continuousAt_symm (e.map_source hx) | theorem | OpenPartialHomeomorph.tendsto_symm | Topology.OpenPartialHomeomorph | Mathlib/Topology/OpenPartialHomeomorph/Continuity.lean | [] | [
"ContinuousAt"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
map_nhds_eq {x} (hx : x ∈ e.source) : map e (𝓝 x) = 𝓝 (e x) | le_antisymm (e.continuousAt hx) <|
le_map_of_right_inverse (e.eventually_right_inverse' hx) (e.tendsto_symm hx) | theorem | OpenPartialHomeomorph.map_nhds_eq | Topology.OpenPartialHomeomorph | Mathlib/Topology/OpenPartialHomeomorph/Continuity.lean | [] | [
"le_antisymm"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
symm_map_nhds_eq {x} (hx : x ∈ e.source) : map e.symm (𝓝 (e x)) = 𝓝 x | (e.symm.map_nhds_eq <| e.map_source hx).trans <| by rw [e.left_inv hx] | theorem | OpenPartialHomeomorph.symm_map_nhds_eq | Topology.OpenPartialHomeomorph | Mathlib/Topology/OpenPartialHomeomorph/Continuity.lean | [] | [
"trans"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
image_mem_nhds {x} (hx : x ∈ e.source) {s : Set X} (hs : s ∈ 𝓝 x) : e '' s ∈ 𝓝 (e x) | e.map_nhds_eq hx ▸ Filter.image_mem_map hs | theorem | OpenPartialHomeomorph.image_mem_nhds | Topology.OpenPartialHomeomorph | Mathlib/Topology/OpenPartialHomeomorph/Continuity.lean | [] | [
"Filter.image_mem_map",
"Set"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
map_nhdsWithin_eq {x} (hx : x ∈ e.source) (s : Set X) :
map e (𝓝[s] x) = 𝓝[e '' (e.source ∩ s)] e x | calc
map e (𝓝[s] x) = map e (𝓝[e.source ∩ s] x) :=
congr_arg (map e) (e.nhdsWithin_source_inter hx _).symm
_ = 𝓝[e '' (e.source ∩ s)] e x :=
(e.leftInvOn.mono inter_subset_left).map_nhdsWithin_eq (e.left_inv hx)
(e.continuousAt_symm (e.map_source hx)).continuousWithinAt
(e.continu... | theorem | OpenPartialHomeomorph.map_nhdsWithin_eq | Topology.OpenPartialHomeomorph | Mathlib/Topology/OpenPartialHomeomorph/Continuity.lean | [] | [
"Set",
"symm"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
map_nhdsWithin_preimage_eq {x} (hx : x ∈ e.source) (s : Set Y) :
map e (𝓝[e ⁻¹' s] x) = 𝓝[s] e x | by
rw [e.map_nhdsWithin_eq hx, e.image_source_inter_eq', e.target_inter_inv_preimage_preimage,
e.nhdsWithin_target_inter (e.map_source hx)] | theorem | OpenPartialHomeomorph.map_nhdsWithin_preimage_eq | Topology.OpenPartialHomeomorph | Mathlib/Topology/OpenPartialHomeomorph/Continuity.lean | [] | [
"Set"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
eventually_nhds {x : X} (p : Y → Prop) (hx : x ∈ e.source) :
(∀ᶠ y in 𝓝 (e x), p y) ↔ ∀ᶠ x in 𝓝 x, p (e x) | Iff.trans (by rw [e.map_nhds_eq hx]) eventually_map | theorem | OpenPartialHomeomorph.eventually_nhds | Topology.OpenPartialHomeomorph | Mathlib/Topology/OpenPartialHomeomorph/Continuity.lean | [] | [] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
eventually_nhds' {x : X} (p : X → Prop) (hx : x ∈ e.source) :
(∀ᶠ y in 𝓝 (e x), p (e.symm y)) ↔ ∀ᶠ x in 𝓝 x, p x | by
rw [e.eventually_nhds _ hx]
refine eventually_congr ((e.eventually_left_inverse hx).mono fun y hy => ?_)
rw [hy] | theorem | OpenPartialHomeomorph.eventually_nhds' | Topology.OpenPartialHomeomorph | Mathlib/Topology/OpenPartialHomeomorph/Continuity.lean | [] | [] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
eventually_nhdsWithin {x : X} (p : Y → Prop) {s : Set X}
(hx : x ∈ e.source) : (∀ᶠ y in 𝓝[e.symm ⁻¹' s] e x, p y) ↔ ∀ᶠ x in 𝓝[s] x, p (e x) | by
refine Iff.trans ?_ eventually_map
rw [e.map_nhdsWithin_eq hx, e.image_source_inter_eq', e.nhdsWithin_target_inter (e.mapsTo hx)] | theorem | OpenPartialHomeomorph.eventually_nhdsWithin | Topology.OpenPartialHomeomorph | Mathlib/Topology/OpenPartialHomeomorph/Continuity.lean | [] | [
"Set"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
eventually_nhdsWithin' {x : X} (p : X → Prop) {s : Set X}
(hx : x ∈ e.source) : (∀ᶠ y in 𝓝[e.symm ⁻¹' s] e x, p (e.symm y)) ↔ ∀ᶠ x in 𝓝[s] x, p x | by
rw [e.eventually_nhdsWithin _ hx]
refine eventually_congr <|
(eventually_nhdsWithin_of_eventually_nhds <| e.eventually_left_inverse hx).mono fun y hy => ?_
rw [hy] | theorem | OpenPartialHomeomorph.eventually_nhdsWithin' | Topology.OpenPartialHomeomorph | Mathlib/Topology/OpenPartialHomeomorph/Continuity.lean | [] | [
"Set",
"eventually_nhdsWithin_of_eventually_nhds"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
preimage_eventuallyEq_target_inter_preimage_inter {e : OpenPartialHomeomorph X Y}
{s : Set X} {t : Set Z} {x : X} {f : X → Z} (hf : ContinuousWithinAt f s x) (hxe : x ∈ e.source)
(ht : t ∈ 𝓝 (f x)) :
e.symm ⁻¹' s =ᶠ[𝓝 (e x)] (e.target ∩ e.symm ⁻¹' (s ∩ f ⁻¹' t) : Set Y) | by
rw [eventuallyEq_set, e.eventually_nhds _ hxe]
filter_upwards [e.open_source.mem_nhds hxe,
mem_nhdsWithin_iff_eventually.mp (hf.preimage_mem_nhdsWithin ht)]
intro y hy hyu
simp_rw [mem_inter_iff, mem_preimage, mem_inter_iff, e.mapsTo hy, true_and, iff_self_and,
e.left_inv hy, iff_true_intro hyu] | theorem | OpenPartialHomeomorph.preimage_eventuallyEq_target_inter_preimage_inter | Topology.OpenPartialHomeomorph | Mathlib/Topology/OpenPartialHomeomorph/Continuity.lean | [] | [
"ContinuousWithinAt",
"OpenPartialHomeomorph",
"Set"
] | 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`). | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
continuousWithinAt_iff_continuousWithinAt_comp_right {f : Y → Z} {s : Set Y} {x : Y}
(h : x ∈ e.target) :
ContinuousWithinAt f s x ↔ ContinuousWithinAt (f ∘ e) (e ⁻¹' s) (e.symm x) | by
simp_rw [ContinuousWithinAt, ← @tendsto_map'_iff _ _ _ _ e,
e.map_nhdsWithin_preimage_eq (e.map_target h), (· ∘ ·), e.right_inv h] | theorem | OpenPartialHomeomorph.continuousWithinAt_iff_continuousWithinAt_comp_right | Topology.OpenPartialHomeomorph | Mathlib/Topology/OpenPartialHomeomorph/Continuity.lean | [] | [
"ContinuousWithinAt",
"Set"
] | Continuity within a set at a point can be read under right composition with a local
homeomorphism, if the point is in its target | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
continuousAt_iff_continuousAt_comp_right {f : Y → Z} {x : Y} (h : x ∈ e.target) :
ContinuousAt f x ↔ ContinuousAt (f ∘ e) (e.symm x) | by
rw [← continuousWithinAt_univ, e.continuousWithinAt_iff_continuousWithinAt_comp_right h,
preimage_univ, continuousWithinAt_univ] | theorem | OpenPartialHomeomorph.continuousAt_iff_continuousAt_comp_right | Topology.OpenPartialHomeomorph | Mathlib/Topology/OpenPartialHomeomorph/Continuity.lean | [] | [
"ContinuousAt",
"continuousWithinAt_univ"
] | Continuity at a point can be read under right composition with an open partial homeomorphism, if
the point is in its target | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
continuousOn_iff_continuousOn_comp_right {f : Y → Z} {s : Set Y} (h : s ⊆ e.target) :
ContinuousOn f s ↔ ContinuousOn (f ∘ e) (e.source ∩ e ⁻¹' s) | by
simp only [← e.symm_image_eq_source_inter_preimage h, ContinuousOn, forall_mem_image]
refine forall₂_congr fun x hx => ?_
rw [e.continuousWithinAt_iff_continuousWithinAt_comp_right (h hx),
e.symm_image_eq_source_inter_preimage h, inter_comm, continuousWithinAt_inter]
exact IsOpen.mem_nhds e.open_source (... | theorem | OpenPartialHomeomorph.continuousOn_iff_continuousOn_comp_right | Topology.OpenPartialHomeomorph | Mathlib/Topology/OpenPartialHomeomorph/Continuity.lean | [] | [
"ContinuousOn",
"IsOpen.mem_nhds",
"Set",
"continuousWithinAt_inter"
] | A function is continuous on a set if and only if its composition with an open partial
homeomorphism on the right is continuous on the corresponding set. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
continuousWithinAt_iff_continuousWithinAt_comp_left {f : Z → X} {s : Set Z} {x : Z}
(hx : f x ∈ e.source) (h : f ⁻¹' e.source ∈ 𝓝[s] x) :
ContinuousWithinAt f s x ↔ ContinuousWithinAt (e ∘ f) s x | by
refine ⟨(e.continuousAt hx).comp_continuousWithinAt, fun fe_cont => ?_⟩
rw [← continuousWithinAt_inter' h] at fe_cont ⊢
have : ContinuousWithinAt (e.symm ∘ e ∘ f) (s ∩ f ⁻¹' e.source) x :=
haveI : ContinuousWithinAt e.symm univ (e (f x)) :=
(e.continuousAt_symm (e.map_source hx)).continuousWithinAt
... | theorem | OpenPartialHomeomorph.continuousWithinAt_iff_continuousWithinAt_comp_left | Topology.OpenPartialHomeomorph | Mathlib/Topology/OpenPartialHomeomorph/Continuity.lean | [] | [
"ContinuousWithinAt",
"ContinuousWithinAt.comp",
"Set",
"continuousWithinAt_inter'"
] | 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 | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
continuousAt_iff_continuousAt_comp_left {f : Z → X} {x : Z} (h : f ⁻¹' e.source ∈ 𝓝 x) :
ContinuousAt f x ↔ ContinuousAt (e ∘ f) x | by
have hx : f x ∈ e.source := (mem_of_mem_nhds h :)
have h' : f ⁻¹' e.source ∈ 𝓝[univ] x := by rwa [nhdsWithin_univ]
rw [← continuousWithinAt_univ, ← continuousWithinAt_univ,
e.continuousWithinAt_iff_continuousWithinAt_comp_left hx h'] | theorem | OpenPartialHomeomorph.continuousAt_iff_continuousAt_comp_left | Topology.OpenPartialHomeomorph | Mathlib/Topology/OpenPartialHomeomorph/Continuity.lean | [] | [
"ContinuousAt",
"continuousWithinAt_univ",
"mem_of_mem_nhds",
"nhdsWithin_univ"
] | Continuity at a point can be read under left composition with an open partial homeomorphism if a
neighborhood of the initial point is sent to the source of the partial homeomorphism | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
continuousOn_iff_continuousOn_comp_left {f : Z → X} {s : Set Z} (h : s ⊆ f ⁻¹' e.source) :
ContinuousOn f s ↔ ContinuousOn (e ∘ f) s | forall₂_congr fun _x hx =>
e.continuousWithinAt_iff_continuousWithinAt_comp_left (h hx)
(mem_of_superset self_mem_nhdsWithin h) | theorem | OpenPartialHomeomorph.continuousOn_iff_continuousOn_comp_left | Topology.OpenPartialHomeomorph | Mathlib/Topology/OpenPartialHomeomorph/Continuity.lean | [] | [
"ContinuousOn",
"Set",
"self_mem_nhdsWithin"
] | A function is continuous on a set if and only if its composition with an open partial
homeomorphism on the left is continuous on the corresponding set. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
continuous_iff_continuous_comp_left {f : Z → X} (h : f ⁻¹' e.source = univ) :
Continuous f ↔ Continuous (e ∘ f) | by
simp only [← continuousOn_univ]
exact e.continuousOn_iff_continuousOn_comp_left (Eq.symm h).subset | theorem | OpenPartialHomeomorph.continuous_iff_continuous_comp_left | Topology.OpenPartialHomeomorph | Mathlib/Topology/OpenPartialHomeomorph/Continuity.lean | [] | [
"Continuous",
"continuousOn_univ"
] | A function is continuous if and only if its composition with an open partial homeomorphism
on the left is continuous and its image is contained in the source. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
OpenPartialHomeomorph (X : Type*) (Y : Type*) [TopologicalSpace X]
[TopologicalSpace Y] extends PartialEquiv X Y where
open_source : IsOpen source
open_target : IsOpen target
continuousOn_toFun : ContinuousOn toFun source
continuousOn_invFun : ContinuousOn invFun target | structure | OpenPartialHomeomorph | Topology.OpenPartialHomeomorph | Mathlib/Topology/OpenPartialHomeomorph/Defs.lean | [] | [
"ContinuousOn",
"IsOpen",
"PartialEquiv",
"TopologicalSpace"
] | Partial homeomorphisms, defined on open subsets of the space | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
toFun' : X → Y | e.toFun | def | OpenPartialHomeomorph.toFun' | Topology.OpenPartialHomeomorph | Mathlib/Topology/OpenPartialHomeomorph/Defs.lean | [] | [] | Coercion of an open 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. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
symm : OpenPartialHomeomorph Y X | where
toPartialEquiv := e.toPartialEquiv.symm
open_source := e.open_target
open_target := e.open_source
continuousOn_toFun := e.continuousOn_invFun
continuousOn_invFun := e.continuousOn_toFun | def | OpenPartialHomeomorph.symm | Topology.OpenPartialHomeomorph | Mathlib/Topology/OpenPartialHomeomorph/Defs.lean | [] | [
"OpenPartialHomeomorph",
"symm"
] | The inverse of an open partial homeomorphism | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
Simps.apply (e : OpenPartialHomeomorph X Y) : X → Y | e | def | OpenPartialHomeomorph.Simps.apply | Topology.OpenPartialHomeomorph | Mathlib/Topology/OpenPartialHomeomorph/Defs.lean | [] | [
"OpenPartialHomeomorph"
] | See Note [custom simps projection]. We need to specify this projection explicitly in this case,
because it is a composition of multiple projections. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
Simps.symm_apply (e : OpenPartialHomeomorph X Y) : Y → X | e.symm
initialize_simps_projections OpenPartialHomeomorph (toFun → apply, invFun → symm_apply) | def | OpenPartialHomeomorph.Simps.symm_apply | Topology.OpenPartialHomeomorph | Mathlib/Topology/OpenPartialHomeomorph/Defs.lean | [] | [
"OpenPartialHomeomorph"
] | See Note [custom simps projection] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
continuousOn : ContinuousOn e e.source | e.continuousOn_toFun | theorem | OpenPartialHomeomorph.continuousOn | Topology.OpenPartialHomeomorph | Mathlib/Topology/OpenPartialHomeomorph/Defs.lean | [] | [
"ContinuousOn"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
continuousOn_symm : ContinuousOn e.symm e.target | e.continuousOn_invFun | theorem | OpenPartialHomeomorph.continuousOn_symm | Topology.OpenPartialHomeomorph | Mathlib/Topology/OpenPartialHomeomorph/Defs.lean | [] | [
"ContinuousOn"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
coe_mk (e : PartialEquiv X Y) (h₁ h₂ h₃ h₄) :
(OpenPartialHomeomorph.mk e h₁ h₂ h₃ h₄ : X → Y) = e | rfl
@[deprecated (since := "2026-05-20")] alias mk_coe := coe_mk | theorem | OpenPartialHomeomorph.coe_mk | Topology.OpenPartialHomeomorph | Mathlib/Topology/OpenPartialHomeomorph/Defs.lean | [] | [
"PartialEquiv"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
coe_mk_symm (e : PartialEquiv X Y) (h₁ h₂ h₃ h₄) :
((OpenPartialHomeomorph.mk e h₁ h₂ h₃ h₄).symm : Y → X) = e.symm | rfl
@[deprecated (since := "2026-05-20")] alias mk_coe_symm := coe_mk_symm | theorem | OpenPartialHomeomorph.coe_mk_symm | Topology.OpenPartialHomeomorph | Mathlib/Topology/OpenPartialHomeomorph/Defs.lean | [] | [
"PartialEquiv",
"symm"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
toPartialEquiv_injective :
Injective (toPartialEquiv : OpenPartialHomeomorph X Y → PartialEquiv X Y) | | ⟨_, _, _, _, _⟩, ⟨_, _, _, _, _⟩, rfl => rfl | theorem | OpenPartialHomeomorph.toPartialEquiv_injective | Topology.OpenPartialHomeomorph | Mathlib/Topology/OpenPartialHomeomorph/Defs.lean | [] | [
"OpenPartialHomeomorph",
"PartialEquiv"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
toFun_eq_coe (e : OpenPartialHomeomorph X Y) : e.toFun = e | rfl | theorem | OpenPartialHomeomorph.toFun_eq_coe | Topology.OpenPartialHomeomorph | Mathlib/Topology/OpenPartialHomeomorph/Defs.lean | [] | [
"OpenPartialHomeomorph"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
invFun_eq_coe (e : OpenPartialHomeomorph X Y) : e.invFun = e.symm | rfl | theorem | OpenPartialHomeomorph.invFun_eq_coe | Topology.OpenPartialHomeomorph | Mathlib/Topology/OpenPartialHomeomorph/Defs.lean | [] | [
"OpenPartialHomeomorph"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
coe_toPartialEquiv : (e.toPartialEquiv : X → Y) = e | rfl
@[deprecated (since := "2026-05-18")] alias coe_coe := coe_toPartialEquiv | theorem | OpenPartialHomeomorph.coe_toPartialEquiv | Topology.OpenPartialHomeomorph | Mathlib/Topology/OpenPartialHomeomorph/Defs.lean | [] | [] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
coe_toPartialEquiv_symm : (e.toPartialEquiv.symm : Y → X) = e.symm | rfl
@[deprecated (since := "2026-05-18")] alias coe_coe_symm := coe_toPartialEquiv_symm | theorem | OpenPartialHomeomorph.coe_toPartialEquiv_symm | Topology.OpenPartialHomeomorph | Mathlib/Topology/OpenPartialHomeomorph/Defs.lean | [] | [] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
map_source {x : X} (h : x ∈ e.source) : e x ∈ e.target | e.map_source' h | theorem | OpenPartialHomeomorph.map_source | Topology.OpenPartialHomeomorph | Mathlib/Topology/OpenPartialHomeomorph/Defs.lean | [] | [] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
map_target {x : Y} (h : x ∈ e.target) : e.symm x ∈ e.source | e.map_target' h | theorem | OpenPartialHomeomorph.map_target | Topology.OpenPartialHomeomorph | Mathlib/Topology/OpenPartialHomeomorph/Defs.lean | [] | [] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
left_inv {x : X} (h : x ∈ e.source) : e.symm (e x) = x | e.left_inv' h | theorem | OpenPartialHomeomorph.left_inv | Topology.OpenPartialHomeomorph | Mathlib/Topology/OpenPartialHomeomorph/Defs.lean | [] | [] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
right_inv {x : Y} (h : x ∈ e.target) : e (e.symm x) = x | e.right_inv' h | theorem | OpenPartialHomeomorph.right_inv | Topology.OpenPartialHomeomorph | Mathlib/Topology/OpenPartialHomeomorph/Defs.lean | [] | [] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
eq_symm_apply {x : X} {y : Y} (hx : x ∈ e.source) (hy : y ∈ e.target) :
x = e.symm y ↔ e x = y | e.toPartialEquiv.eq_symm_apply hx hy | theorem | OpenPartialHomeomorph.eq_symm_apply | Topology.OpenPartialHomeomorph | Mathlib/Topology/OpenPartialHomeomorph/Defs.lean | [] | [] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
mapsTo_symm : MapsTo e.symm e.target e.source | e.symm.mapsTo
@[deprecated (since := "2026-05-28")] alias symm_mapsTo := OpenPartialHomeomorph.mapsTo_symm | theorem | OpenPartialHomeomorph.mapsTo_symm | Topology.OpenPartialHomeomorph | Mathlib/Topology/OpenPartialHomeomorph/Defs.lean | [] | [] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
_root_.Homeomorph.toOpenPartialHomeomorphOfImageEq (e : X ≃ₜ Y) (s : Set X) (hs : IsOpen s)
(t : Set Y) (h : e '' s = t) : OpenPartialHomeomorph X Y | where
toPartialEquiv := e.toPartialEquivOfImageEq s t h
open_source := hs
open_target := by simpa [← h]
continuousOn_toFun := e.continuous.continuousOn
continuousOn_invFun := e.symm.continuous.continuousOn | def | Homeomorph.toOpenPartialHomeomorphOfImageEq | Topology.OpenPartialHomeomorph | Mathlib/Topology/OpenPartialHomeomorph/Defs.lean | [] | [
"IsOpen",
"OpenPartialHomeomorph",
"Set"
] | Interpret a `Homeomorph` as an `OpenPartialHomeomorph` by restricting it
to an open set `s` in the domain and to `t` in the codomain. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
_root_.Homeomorph.toOpenPartialHomeomorph (e : X ≃ₜ Y) : OpenPartialHomeomorph X Y | e.toOpenPartialHomeomorphOfImageEq univ isOpen_univ univ <|
by rw [image_univ, e.surjective.range_eq] | def | Homeomorph.toOpenPartialHomeomorph | Topology.OpenPartialHomeomorph | Mathlib/Topology/OpenPartialHomeomorph/Defs.lean | [] | [
"OpenPartialHomeomorph",
"isOpen_univ"
] | A homeomorphism induces an open partial homeomorphism on the whole space | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
replacePartialEquiv (e : OpenPartialHomeomorph X Y) (e' : PartialEquiv X Y)
(h : e.toPartialEquiv = e') : OpenPartialHomeomorph X Y | where
toPartialEquiv := e'
open_source := h ▸ e.open_source
open_target := h ▸ e.open_target
continuousOn_toFun := h ▸ e.continuousOn_toFun
continuousOn_invFun := h ▸ e.continuousOn_invFun
@[deprecated (since := "2026-05-19")] alias replaceEquiv := replacePartialEquiv | def | OpenPartialHomeomorph.replacePartialEquiv | Topology.OpenPartialHomeomorph | Mathlib/Topology/OpenPartialHomeomorph/Defs.lean | [] | [
"OpenPartialHomeomorph",
"PartialEquiv"
] | Replace `toPartialEquiv` field to provide better definitional equalities. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
replacePartialEquiv_eq_self (e' : PartialEquiv X Y)
(h : e.toPartialEquiv = e') : e.replacePartialEquiv e' h = e | by
cases e
subst e'
rfl
@[deprecated (since := "2026-05-20")] alias replaceEquiv_eq_self := replacePartialEquiv_eq_self | theorem | OpenPartialHomeomorph.replacePartialEquiv_eq_self | Topology.OpenPartialHomeomorph | Mathlib/Topology/OpenPartialHomeomorph/Defs.lean | [] | [
"PartialEquiv"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
ext (e' : OpenPartialHomeomorph X Y) (h : ∀ x, e x = e' x)
(hinv : ∀ x, e.symm x = e'.symm x) (hs : e.source = e'.source) : e = e' | toPartialEquiv_injective (PartialEquiv.ext h hinv hs) | theorem | OpenPartialHomeomorph.ext | Topology.OpenPartialHomeomorph | Mathlib/Topology/OpenPartialHomeomorph/Defs.lean | [] | [
"OpenPartialHomeomorph",
"PartialEquiv.ext"
] | Two open 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`. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
symm_bijective : Function.Bijective
(OpenPartialHomeomorph.symm : OpenPartialHomeomorph X Y → OpenPartialHomeomorph Y X) | Function.bijective_iff_has_inverse.mpr ⟨_, symm_symm, symm_symm⟩ | theorem | OpenPartialHomeomorph.symm_bijective | Topology.OpenPartialHomeomorph | Mathlib/Topology/OpenPartialHomeomorph/Defs.lean | [] | [
"Function.Bijective",
"OpenPartialHomeomorph",
"OpenPartialHomeomorph.symm"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
IsImage (s : Set X) (t : Set Y) : Prop | ∀ ⦃x⦄, x ∈ e.source → (e x ∈ t ↔ x ∈ s) | def | OpenPartialHomeomorph.IsImage | Topology.OpenPartialHomeomorph | Mathlib/Topology/OpenPartialHomeomorph/IsImage.lean | [] | [
"Set"
] | We say that `t : Set Y` is an image of `s : Set X` under an open 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). | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
toPartialEquiv (h : e.IsImage s t) : e.toPartialEquiv.IsImage s t | h | theorem | OpenPartialHomeomorph.IsImage.toPartialEquiv | Topology.OpenPartialHomeomorph | Mathlib/Topology/OpenPartialHomeomorph/IsImage.lean | [] | [] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
symm (h : e.IsImage s t) : e.symm.IsImage t s | h.toPartialEquiv.symm | theorem | OpenPartialHomeomorph.IsImage.symm | Topology.OpenPartialHomeomorph | Mathlib/Topology/OpenPartialHomeomorph/IsImage.lean | [] | [
"symm"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
symm_apply_mem_iff (h : e.IsImage s t) (hy : y ∈ e.target) : e.symm y ∈ s ↔ y ∈ t | h.symm hy | theorem | OpenPartialHomeomorph.IsImage.symm_apply_mem_iff | Topology.OpenPartialHomeomorph | Mathlib/Topology/OpenPartialHomeomorph/IsImage.lean | [] | [] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
mapsTo (h : e.IsImage s t) : MapsTo e (e.source ∩ s) (e.target ∩ t) | h.toPartialEquiv.mapsTo | theorem | OpenPartialHomeomorph.IsImage.mapsTo | Topology.OpenPartialHomeomorph | Mathlib/Topology/OpenPartialHomeomorph/IsImage.lean | [] | [] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
image_eq (h : e.IsImage s t) : e '' (e.source ∩ s) = e.target ∩ t | h.toPartialEquiv.image_eq | theorem | OpenPartialHomeomorph.IsImage.image_eq | Topology.OpenPartialHomeomorph | Mathlib/Topology/OpenPartialHomeomorph/IsImage.lean | [] | [] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
iff_preimage_eq : e.IsImage s t ↔ e.source ∩ e ⁻¹' t = e.source ∩ s | PartialEquiv.IsImage.iff_preimage_eq
alias ⟨preimage_eq, of_preimage_eq⟩ := iff_preimage_eq | theorem | OpenPartialHomeomorph.IsImage.iff_preimage_eq | Topology.OpenPartialHomeomorph | Mathlib/Topology/OpenPartialHomeomorph/IsImage.lean | [] | [
"PartialEquiv.IsImage.iff_preimage_eq"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
iff_symm_preimage_eq' :
e.IsImage s t ↔ e.target ∩ e.symm ⁻¹' (e.source ∩ s) = e.target ∩ t | by
rw [iff_symm_preimage_eq, ← image_source_inter_eq, ← image_source_inter_eq']
alias ⟨symm_preimage_eq', of_symm_preimage_eq'⟩ := iff_symm_preimage_eq' | theorem | OpenPartialHomeomorph.IsImage.iff_symm_preimage_eq' | Topology.OpenPartialHomeomorph | Mathlib/Topology/OpenPartialHomeomorph/IsImage.lean | [] | [] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
iff_preimage_eq' : e.IsImage s t ↔ e.source ∩ e ⁻¹' (e.target ∩ t) = e.source ∩ s | symm_iff.symm.trans iff_symm_preimage_eq'
alias ⟨preimage_eq', of_preimage_eq'⟩ := iff_preimage_eq' | theorem | OpenPartialHomeomorph.IsImage.iff_preimage_eq' | Topology.OpenPartialHomeomorph | Mathlib/Topology/OpenPartialHomeomorph/IsImage.lean | [] | [] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
of_image_eq (h : e '' (e.source ∩ s) = e.target ∩ t) : e.IsImage s t | PartialEquiv.IsImage.of_image_eq h | theorem | OpenPartialHomeomorph.IsImage.of_image_eq | Topology.OpenPartialHomeomorph | Mathlib/Topology/OpenPartialHomeomorph/IsImage.lean | [] | [
"PartialEquiv.IsImage.of_image_eq"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
of_symm_image_eq (h : e.symm '' (e.target ∩ t) = e.source ∩ s) : e.IsImage s t | PartialEquiv.IsImage.of_symm_image_eq h | theorem | OpenPartialHomeomorph.IsImage.of_symm_image_eq | Topology.OpenPartialHomeomorph | Mathlib/Topology/OpenPartialHomeomorph/IsImage.lean | [] | [
"PartialEquiv.IsImage.of_symm_image_eq"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
compl (h : e.IsImage s t) : e.IsImage sᶜ tᶜ | fun _ hx => (h hx).not | theorem | OpenPartialHomeomorph.IsImage.compl | Topology.OpenPartialHomeomorph | Mathlib/Topology/OpenPartialHomeomorph/IsImage.lean | [] | [] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
inter {s' t'} (h : e.IsImage s t) (h' : e.IsImage s' t') :
e.IsImage (s ∩ s') (t ∩ t') | fun _ hx => (h hx).and (h' hx) | theorem | OpenPartialHomeomorph.IsImage.inter | Topology.OpenPartialHomeomorph | Mathlib/Topology/OpenPartialHomeomorph/IsImage.lean | [] | [] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
union {s' t'} (h : e.IsImage s t) (h' : e.IsImage s' t') :
e.IsImage (s ∪ s') (t ∪ t') | fun _ hx => (h hx).or (h' hx) | theorem | OpenPartialHomeomorph.IsImage.union | Topology.OpenPartialHomeomorph | Mathlib/Topology/OpenPartialHomeomorph/IsImage.lean | [] | [] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
leftInvOn_piecewise {e' : OpenPartialHomeomorph X Y} [∀ i, Decidable (i ∈ s)]
[∀ i, Decidable (i ∈ t)] (h : e.IsImage s t) (h' : e'.IsImage s t) :
LeftInvOn (t.piecewise e.symm e'.symm) (s.piecewise e e') (s.ite e.source e'.source) | h.toPartialEquiv.leftInvOn_piecewise h' | theorem | OpenPartialHomeomorph.IsImage.leftInvOn_piecewise | Topology.OpenPartialHomeomorph | Mathlib/Topology/OpenPartialHomeomorph/IsImage.lean | [] | [
"OpenPartialHomeomorph"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
inter_eq_of_inter_eq_of_eqOn {e' : OpenPartialHomeomorph X Y} (h : e.IsImage s t)
(h' : e'.IsImage s t) (hs : e.source ∩ s = e'.source ∩ s) (Heq : EqOn e e' (e.source ∩ s)) :
e.target ∩ t = e'.target ∩ t | h.toPartialEquiv.inter_eq_of_inter_eq_of_eqOn h' hs Heq | theorem | OpenPartialHomeomorph.IsImage.inter_eq_of_inter_eq_of_eqOn | Topology.OpenPartialHomeomorph | Mathlib/Topology/OpenPartialHomeomorph/IsImage.lean | [] | [
"OpenPartialHomeomorph"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
symm_eqOn_of_inter_eq_of_eqOn {e' : OpenPartialHomeomorph X Y} (h : e.IsImage s t)
(hs : e.source ∩ s = e'.source ∩ s) (Heq : EqOn e e' (e.source ∩ s)) :
EqOn e.symm e'.symm (e.target ∩ t) | h.toPartialEquiv.symm_eq_on_of_inter_eq_of_eqOn hs Heq | theorem | OpenPartialHomeomorph.IsImage.symm_eqOn_of_inter_eq_of_eqOn | Topology.OpenPartialHomeomorph | Mathlib/Topology/OpenPartialHomeomorph/IsImage.lean | [] | [
"OpenPartialHomeomorph"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
map_nhdsWithin_eq (h : e.IsImage s t) (hx : x ∈ e.source) : map e (𝓝[s] x) = 𝓝[t] e x | by
rw [e.map_nhdsWithin_eq hx, h.image_eq, e.nhdsWithin_target_inter (e.map_source hx)] | theorem | OpenPartialHomeomorph.IsImage.map_nhdsWithin_eq | Topology.OpenPartialHomeomorph | Mathlib/Topology/OpenPartialHomeomorph/IsImage.lean | [] | [] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
closure (h : e.IsImage s t) : e.IsImage (closure s) (closure t) | fun x hx => by
simp only [mem_closure_iff_nhdsWithin_neBot, ← h.map_nhdsWithin_eq hx, map_neBot_iff] | theorem | OpenPartialHomeomorph.IsImage.closure | Topology.OpenPartialHomeomorph | Mathlib/Topology/OpenPartialHomeomorph/IsImage.lean | [] | [
"closure",
"mem_closure_iff_nhdsWithin_neBot"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
interior (h : e.IsImage s t) : e.IsImage (interior s) (interior t) | by
simpa only [closure_compl, compl_compl] using h.compl.closure.compl | theorem | OpenPartialHomeomorph.IsImage.interior | Topology.OpenPartialHomeomorph | Mathlib/Topology/OpenPartialHomeomorph/IsImage.lean | [] | [
"closure_compl",
"compl_compl",
"interior"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
frontier (h : e.IsImage s t) : e.IsImage (frontier s) (frontier t) | h.closure.diff h.interior | theorem | OpenPartialHomeomorph.IsImage.frontier | Topology.OpenPartialHomeomorph | Mathlib/Topology/OpenPartialHomeomorph/IsImage.lean | [] | [
"frontier"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
isOpen_iff (h : e.IsImage s t) : IsOpen (e.source ∩ s) ↔ IsOpen (e.target ∩ t) | ⟨fun hs => h.symm_preimage_eq' ▸ e.symm.isOpen_inter_preimage hs, fun hs =>
h.preimage_eq' ▸ e.isOpen_inter_preimage hs⟩ | theorem | OpenPartialHomeomorph.IsImage.isOpen_iff | Topology.OpenPartialHomeomorph | Mathlib/Topology/OpenPartialHomeomorph/IsImage.lean | [] | [
"IsOpen"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
restr (h : e.IsImage s t) (hs : IsOpen (e.source ∩ s)) : OpenPartialHomeomorph X Y | where
toPartialEquiv := h.toPartialEquiv.restr
open_source := hs
open_target := h.isOpen_iff.1 hs
continuousOn_toFun := e.continuousOn.mono inter_subset_left
continuousOn_invFun := e.symm.continuousOn.mono inter_subset_left | def | OpenPartialHomeomorph.IsImage.restr | Topology.OpenPartialHomeomorph | Mathlib/Topology/OpenPartialHomeomorph/IsImage.lean | [] | [
"IsOpen",
"OpenPartialHomeomorph"
] | Restrict an `OpenPartialHomeomorph` to a pair of corresponding open sets. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
isImage_source_target : e.IsImage e.source e.target | e.toPartialEquiv.isImage_source_target | theorem | OpenPartialHomeomorph.isImage_source_target | Topology.OpenPartialHomeomorph | Mathlib/Topology/OpenPartialHomeomorph/IsImage.lean | [] | [] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
isImage_source_target_of_disjoint (e' : OpenPartialHomeomorph X Y)
(hs : Disjoint e.source e'.source) (ht : Disjoint e.target e'.target) :
e.IsImage e'.source e'.target | e.toPartialEquiv.isImage_source_target_of_disjoint e'.toPartialEquiv hs ht | theorem | OpenPartialHomeomorph.isImage_source_target_of_disjoint | Topology.OpenPartialHomeomorph | Mathlib/Topology/OpenPartialHomeomorph/IsImage.lean | [] | [
"Disjoint",
"OpenPartialHomeomorph"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
preimage_interior (s : Set Y) :
e.source ∩ e ⁻¹' interior s = e.source ∩ interior (e ⁻¹' s) | (IsImage.of_preimage_eq rfl).interior.preimage_eq | theorem | OpenPartialHomeomorph.preimage_interior | Topology.OpenPartialHomeomorph | Mathlib/Topology/OpenPartialHomeomorph/IsImage.lean | [] | [
"Set",
"interior"
] | Preimage of interior or interior of preimage coincide for open partial homeomorphisms,
when restricted to the source. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
preimage_closure (s : Set Y) : e.source ∩ e ⁻¹' closure s = e.source ∩ closure (e ⁻¹' s) | (IsImage.of_preimage_eq rfl).closure.preimage_eq | theorem | OpenPartialHomeomorph.preimage_closure | Topology.OpenPartialHomeomorph | Mathlib/Topology/OpenPartialHomeomorph/IsImage.lean | [] | [
"Set",
"closure"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
preimage_frontier (s : Set Y) :
e.source ∩ e ⁻¹' frontier s = e.source ∩ frontier (e ⁻¹' s) | (IsImage.of_preimage_eq rfl).frontier.preimage_eq | theorem | OpenPartialHomeomorph.preimage_frontier | Topology.OpenPartialHomeomorph | Mathlib/Topology/OpenPartialHomeomorph/IsImage.lean | [] | [
"Set",
"frontier"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
restrOpen (s : Set X) (hs : IsOpen s) : OpenPartialHomeomorph X Y | (@IsImage.of_symm_preimage_eq X Y _ _ e s (e.symm ⁻¹' s) rfl).restr
(IsOpen.inter e.open_source hs) | def | OpenPartialHomeomorph.restrOpen | Topology.OpenPartialHomeomorph | Mathlib/Topology/OpenPartialHomeomorph/IsImage.lean | [] | [
"IsOpen",
"IsOpen.inter",
"OpenPartialHomeomorph",
"Set"
] | Restricting an open 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`. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
restrOpen_toPartialEquiv (s : Set X) (hs : IsOpen s) :
(e.restrOpen s hs).toPartialEquiv = e.toPartialEquiv.restr s | rfl | theorem | OpenPartialHomeomorph.restrOpen_toPartialEquiv | Topology.OpenPartialHomeomorph | Mathlib/Topology/OpenPartialHomeomorph/IsImage.lean | [] | [
"IsOpen",
"Set"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
restrOpen_source (s : Set X) (hs : IsOpen s) : (e.restrOpen s hs).source = e.source ∩ s | rfl | theorem | OpenPartialHomeomorph.restrOpen_source | Topology.OpenPartialHomeomorph | Mathlib/Topology/OpenPartialHomeomorph/IsImage.lean | [] | [
"IsOpen",
"Set"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
coe_restrOpen {s : Set X} (hs : IsOpen s) : ⇑(e.restrOpen s hs) = e | rfl | theorem | OpenPartialHomeomorph.coe_restrOpen | Topology.OpenPartialHomeomorph | Mathlib/Topology/OpenPartialHomeomorph/IsImage.lean | [] | [
"IsOpen",
"Set"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
coe_restrOpen_symm {s : Set X} (hs : IsOpen s) : ⇑(e.restrOpen s hs).symm = e.symm | rfl | theorem | OpenPartialHomeomorph.coe_restrOpen_symm | Topology.OpenPartialHomeomorph | Mathlib/Topology/OpenPartialHomeomorph/IsImage.lean | [] | [
"IsOpen",
"Set",
"symm"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
restr (s : Set X) : OpenPartialHomeomorph X Y | e.restrOpen (interior s) isOpen_interior | def | OpenPartialHomeomorph.restr | Topology.OpenPartialHomeomorph | Mathlib/Topology/OpenPartialHomeomorph/IsImage.lean | [] | [
"OpenPartialHomeomorph",
"Set",
"interior",
"isOpen_interior"
] | Restricting an open 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 open partial homeomorphisms
are by definition defined on open sets. In applications where `s` is open, this coincides with the
restriction of partial ... | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
restr_toPartialEquiv (s : Set X) :
(e.restr s).toPartialEquiv = e.toPartialEquiv.restr (interior s) | rfl | theorem | OpenPartialHomeomorph.restr_toPartialEquiv | Topology.OpenPartialHomeomorph | Mathlib/Topology/OpenPartialHomeomorph/IsImage.lean | [] | [
"Set",
"interior"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
restr_source' (s : Set X) (hs : IsOpen s) : (e.restr s).source = e.source ∩ s | by
grind | theorem | OpenPartialHomeomorph.restr_source' | Topology.OpenPartialHomeomorph | Mathlib/Topology/OpenPartialHomeomorph/IsImage.lean | [] | [
"IsOpen",
"Set"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
restr_toPartialEquiv' (s : Set X) (hs : IsOpen s) :
(e.restr s).toPartialEquiv = e.toPartialEquiv.restr s | by
rw [e.restr_toPartialEquiv, hs.interior_eq] | theorem | OpenPartialHomeomorph.restr_toPartialEquiv' | Topology.OpenPartialHomeomorph | Mathlib/Topology/OpenPartialHomeomorph/IsImage.lean | [] | [
"IsOpen",
"Set"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
restr_eq_of_source_subset {e : OpenPartialHomeomorph X Y} {s : Set X} (h : e.source ⊆ s) :
e.restr s = e | toPartialEquiv_injective <| PartialEquiv.restr_eq_of_source_subset <|
interior_maximal h e.open_source | theorem | OpenPartialHomeomorph.restr_eq_of_source_subset | Topology.OpenPartialHomeomorph | Mathlib/Topology/OpenPartialHomeomorph/IsImage.lean | [] | [
"OpenPartialHomeomorph",
"PartialEquiv.restr_eq_of_source_subset",
"Set",
"interior_maximal"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
restr_univ {e : OpenPartialHomeomorph X Y} : e.restr univ = e | restr_eq_of_source_subset (subset_univ _) | theorem | OpenPartialHomeomorph.restr_univ | Topology.OpenPartialHomeomorph | Mathlib/Topology/OpenPartialHomeomorph/IsImage.lean | [] | [
"OpenPartialHomeomorph"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
restr_source_inter (s : Set X) : e.restr (e.source ∩ s) = e.restr s | by
refine OpenPartialHomeomorph.ext _ _ (fun x => rfl) (fun x => rfl) ?_
simp [e.open_source.interior_eq, ← inter_assoc] | theorem | OpenPartialHomeomorph.restr_source_inter | Topology.OpenPartialHomeomorph | Mathlib/Topology/OpenPartialHomeomorph/IsImage.lean | [] | [
"OpenPartialHomeomorph.ext",
"Set"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
ofSet (s : Set X) (hs : IsOpen s) : OpenPartialHomeomorph X X | where
toPartialEquiv := PartialEquiv.ofSet s
open_source := hs
open_target := hs
continuousOn_toFun := continuous_id.continuousOn
continuousOn_invFun := continuous_id.continuousOn | def | OpenPartialHomeomorph.ofSet | Topology.OpenPartialHomeomorph | Mathlib/Topology/OpenPartialHomeomorph/IsImage.lean | [] | [
"IsOpen",
"OpenPartialHomeomorph",
"PartialEquiv.ofSet",
"Set"
] | The identity partial equivalence on a set `s` | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
ofSet_toPartialEquiv : (ofSet s hs).toPartialEquiv = PartialEquiv.ofSet s | rfl | theorem | OpenPartialHomeomorph.ofSet_toPartialEquiv | Topology.OpenPartialHomeomorph | Mathlib/Topology/OpenPartialHomeomorph/IsImage.lean | [] | [
"PartialEquiv.ofSet"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
ofSet_symm : (ofSet s hs).symm = ofSet s hs | rfl | theorem | OpenPartialHomeomorph.ofSet_symm | Topology.OpenPartialHomeomorph | Mathlib/Topology/OpenPartialHomeomorph/IsImage.lean | [] | [
"symm"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
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