fact stringlengths 6 3.84k | type stringclasses 11
values | library stringclasses 32
values | imports listlengths 1 14 | filename stringlengths 20 95 | symbolic_name stringlengths 1 90 | docstring stringlengths 7 20k ⌀ |
|---|---|---|---|---|---|---|
RelCWComplex.cellFrontier_subset_finite_openCell [RelCWComplex C D] (n : ℕ) (i : cell C n) :
∃ I : Π m, Finset (cell C m),
cellFrontier n i ⊆ D ∪ (⋃ (m < n) (j ∈ I m), openCell m j) := by
induction n using Nat.case_strong_induction_on with
| hz => simp [cellFrontier_zero_eq_empty]
| hi n hn =>
classic... | lemma | Topology | [
"Mathlib.Analysis.Normed.Module.RCLike.Real",
"Mathlib.Data.ENat.Basic",
"Mathlib.Logic.Equiv.PartialEquiv",
"Mathlib.Topology.MetricSpace.ProperSpace.Real"
] | Mathlib/Topology/CWComplex/Classical/Basic.lean | RelCWComplex.cellFrontier_subset_finite_openCell | A version of `cellFrontier_subset_base_union_finite_closedCell` using open cells:
The boundary of a cell is contained in a finite union of open cells of a lower dimension. |
CWComplex.cellFrontier_subset_finite_openCell [CWComplex C] (n : ℕ) (i : cell C n) :
∃ I : Π m, Finset (cell C m),
cellFrontier n i ⊆ ⋃ (m < n) (j ∈ I m), openCell m j := by
simpa using RelCWComplex.cellFrontier_subset_finite_openCell n i | lemma | Topology | [
"Mathlib.Analysis.Normed.Module.RCLike.Real",
"Mathlib.Data.ENat.Basic",
"Mathlib.Logic.Equiv.PartialEquiv",
"Mathlib.Topology.MetricSpace.ProperSpace.Real"
] | Mathlib/Topology/CWComplex/Classical/Basic.lean | CWComplex.cellFrontier_subset_finite_openCell | A version of `cellFrontier_subset_finite_closedCell` using open cells: The boundary of a cell is
contained in a finite union of open cells of a lower dimension. |
Subcomplex (C : Set X) {D : Set X} [RelCWComplex C D] where
/-- The underlying set of the subcomplex. -/
carrier : Set X
/-- The indexing set of cells of the subcomplex. -/
I : Π n, Set (cell C n)
/-- A subcomplex is closed. -/
closed' : IsClosed carrier
/-- The union of all open cells of the subcomplex e... | structure | Topology | [
"Mathlib.Analysis.Normed.Module.RCLike.Real",
"Mathlib.Data.ENat.Basic",
"Mathlib.Logic.Equiv.PartialEquiv",
"Mathlib.Topology.MetricSpace.ProperSpace.Real"
] | Mathlib/Topology/CWComplex/Classical/Basic.lean | Subcomplex | A subcomplex is a closed subspace of a CW complex that is the union of open cells of the
CW complex. |
mem_carrier {E : Subcomplex C} {x : X} : x ∈ E.carrier ↔ x ∈ (E : Set X) := Iff.rfl | lemma | Topology | [
"Mathlib.Analysis.Normed.Module.RCLike.Real",
"Mathlib.Data.ENat.Basic",
"Mathlib.Logic.Equiv.PartialEquiv",
"Mathlib.Topology.MetricSpace.ProperSpace.Real"
] | Mathlib/Topology/CWComplex/Classical/Basic.lean | mem_carrier | null |
coe_eq_carrier {E : Subcomplex C} : (E : Set X) = E.carrier := rfl
@[ext] lemma ext {E F : Subcomplex C} (h : ∀ x, x ∈ E ↔ x ∈ F) : E = F :=
SetLike.ext h | lemma | Topology | [
"Mathlib.Analysis.Normed.Module.RCLike.Real",
"Mathlib.Data.ENat.Basic",
"Mathlib.Logic.Equiv.PartialEquiv",
"Mathlib.Topology.MetricSpace.ProperSpace.Real"
] | Mathlib/Topology/CWComplex/Classical/Basic.lean | coe_eq_carrier | null |
eq_iff (E F : Subcomplex C) : E = F ↔ (E : Set X) = F :=
SetLike.coe_injective.eq_iff.symm | lemma | Topology | [
"Mathlib.Analysis.Normed.Module.RCLike.Real",
"Mathlib.Data.ENat.Basic",
"Mathlib.Logic.Equiv.PartialEquiv",
"Mathlib.Topology.MetricSpace.ProperSpace.Real"
] | Mathlib/Topology/CWComplex/Classical/Basic.lean | eq_iff | null |
protected copy (E : Subcomplex C) (F : Set X) (hF : F = E) (J : (n : ℕ) → Set (cell C n))
(hJ : J = E.I) : Subcomplex C :=
{ carrier := F
I := J
closed' := hF.symm ▸ E.closed'
union' := hF.symm ▸ hJ ▸ E.union' }
@[simp] lemma coe_copy (E : Subcomplex C) (F : Set X) (hF : F = E) (J : (n : ℕ) → Set (cel... | def | Topology | [
"Mathlib.Analysis.Normed.Module.RCLike.Real",
"Mathlib.Data.ENat.Basic",
"Mathlib.Logic.Equiv.PartialEquiv",
"Mathlib.Topology.MetricSpace.ProperSpace.Real"
] | Mathlib/Topology/CWComplex/Classical/Basic.lean | copy | Copy of a `Subcomplex` with a new `carrier` equal to the old one. Useful to fix definitional
equalities. |
copy_eq (E : Subcomplex C) (F : Set X) (hF : F = E) (J : (n : ℕ) → Set (cell C n))
(hJ : J = E.I) : E.copy F hF J hJ = E :=
SetLike.coe_injective hF | lemma | Topology | [
"Mathlib.Analysis.Normed.Module.RCLike.Real",
"Mathlib.Data.ENat.Basic",
"Mathlib.Logic.Equiv.PartialEquiv",
"Mathlib.Topology.MetricSpace.ProperSpace.Real"
] | Mathlib/Topology/CWComplex/Classical/Basic.lean | copy_eq | null |
union (E : Subcomplex C) :
D ∪ ⋃ (n : ℕ) (j : E.I n), openCell (C := C) n j.1 = E := by
rw [E.union']
rfl | lemma | Topology | [
"Mathlib.Analysis.Normed.Module.RCLike.Real",
"Mathlib.Data.ENat.Basic",
"Mathlib.Logic.Equiv.PartialEquiv",
"Mathlib.Topology.MetricSpace.ProperSpace.Real"
] | Mathlib/Topology/CWComplex/Classical/Basic.lean | union | null |
closed (E : Subcomplex C) : IsClosed (E : Set X) := E.closed' | lemma | Topology | [
"Mathlib.Analysis.Normed.Module.RCLike.Real",
"Mathlib.Data.ENat.Basic",
"Mathlib.Logic.Equiv.PartialEquiv",
"Mathlib.Topology.MetricSpace.ProperSpace.Real"
] | Mathlib/Topology/CWComplex/Classical/Basic.lean | closed | null |
CWComplex.Subcomplex.union {C : Set X} [CWComplex C] {E : Subcomplex C} :
⋃ (n : ℕ) (j : E.I n), openCell (C := C) n j = E := by
have := RelCWComplex.Subcomplex.union E (C := C)
rw [empty_union] at this
exact this | lemma | Topology | [
"Mathlib.Analysis.Normed.Module.RCLike.Real",
"Mathlib.Data.ENat.Basic",
"Mathlib.Logic.Equiv.PartialEquiv",
"Mathlib.Topology.MetricSpace.ProperSpace.Real"
] | Mathlib/Topology/CWComplex/Classical/Basic.lean | CWComplex.Subcomplex.union | null |
@[simps -isSimp]
RelCWComplex.Subcomplex.mk' [T2Space X] (C : Set X) {D : Set X} [RelCWComplex C D]
(E : Set X) (I : Π n, Set (cell C n))
(closedCell_subset : ∀ (n : ℕ) (i : I n), closedCell (C := C) n i ⊆ E)
(union : D ∪ ⋃ (n : ℕ) (j : I n), openCell (C := C) n j = E) : Subcomplex C where
carrier := E
... | def | Topology | [
"Mathlib.Analysis.Normed.Module.RCLike.Real",
"Mathlib.Data.ENat.Basic",
"Mathlib.Logic.Equiv.PartialEquiv",
"Mathlib.Topology.MetricSpace.ProperSpace.Real"
] | Mathlib/Topology/CWComplex/Classical/Basic.lean | RelCWComplex.Subcomplex.mk' | An alternative version of `Subcomplex.mk`: Instead of requiring that `E` is closed it requires
that for every cell of the subcomplex the corresponding closed cell is a subset of `E`. |
@[simps! -isSimp]
CWComplex.Subcomplex.mk' [T2Space X] (C : Set X) [CWComplex C] (E : Set X)
(I : Π n, Set (cell C n))
(closedCell_subset : ∀ (n : ℕ) (i : I n), closedCell (C := C) n i ⊆ E)
(union : ⋃ (n : ℕ) (j : I n), openCell (C := C) n j = E) : Subcomplex C :=
RelCWComplex.Subcomplex.mk' C E I closedC... | def | Topology | [
"Mathlib.Analysis.Normed.Module.RCLike.Real",
"Mathlib.Data.ENat.Basic",
"Mathlib.Logic.Equiv.PartialEquiv",
"Mathlib.Topology.MetricSpace.ProperSpace.Real"
] | Mathlib/Topology/CWComplex/Classical/Basic.lean | CWComplex.Subcomplex.mk' | An alternative version of `Subcomplex.mk`: Instead of requiring that `E` is closed it requires
that for every cell of the subcomplex the corresponding closed cell is a subset of `E`. |
@[simps -isSimp]
RelCWComplex.Subcomplex.mk'' [T2Space X] (C : Set X) {D : Set X} [RelCWComplex C D] (E : Set X)
(I : Π n, Set (cell C n)) [RelCWComplex E D]
(union : D ∪ ⋃ (n : ℕ) (j : I n), openCell (C := C) n j = E) : Subcomplex C where
carrier := E
I := I
closed' := isClosed
union' := union | def | Topology | [
"Mathlib.Analysis.Normed.Module.RCLike.Real",
"Mathlib.Data.ENat.Basic",
"Mathlib.Logic.Equiv.PartialEquiv",
"Mathlib.Topology.MetricSpace.ProperSpace.Real"
] | Mathlib/Topology/CWComplex/Classical/Basic.lean | RelCWComplex.Subcomplex.mk'' | An alternative version of `Subcomplex.mk`: Instead of requiring that `E` is closed it requires
that `E` is a CW-complex. |
@[simps -isSimp]
CWComplex.Subcomplex.mk'' [T2Space X] (C : Set X) [h : CWComplex C] (E : Set X)
(I : Π n, Set (cell C n)) [CWComplex E]
(union : ⋃ (n : ℕ) (j : I n), openCell (C := C) n j = E) :
Subcomplex C where
carrier := E
I := I
closed' := RelCWComplex.isClosed
union' := by
rw [empty_union... | def | Topology | [
"Mathlib.Analysis.Normed.Module.RCLike.Real",
"Mathlib.Data.ENat.Basic",
"Mathlib.Logic.Equiv.PartialEquiv",
"Mathlib.Topology.MetricSpace.ProperSpace.Real"
] | Mathlib/Topology/CWComplex/Classical/Basic.lean | CWComplex.Subcomplex.mk'' | An alternative version of `Subcomplex.mk`: Instead of requiring that `E` is closed it requires
that `E` is a CW-complex. |
RelCWComplex.Subcomplex.subset_complex {C D : Set X} [RelCWComplex C D] (E : Subcomplex C) :
↑E ⊆ C := by
simp_rw [← union, ← RelCWComplex.union_iUnion_openCell_eq_complex]
exact union_subset_union_right _ (iUnion_mono fun _ ↦ iUnion_mono' fun j ↦ ⟨j, subset_rfl⟩) | lemma | Topology | [
"Mathlib.Analysis.Normed.Module.RCLike.Real",
"Mathlib.Data.ENat.Basic",
"Mathlib.Logic.Equiv.PartialEquiv",
"Mathlib.Topology.MetricSpace.ProperSpace.Real"
] | Mathlib/Topology/CWComplex/Classical/Basic.lean | RelCWComplex.Subcomplex.subset_complex | null |
RelCWComplex.Subcomplex.base_subset {C D : Set X} [RelCWComplex C D] (E : Subcomplex C) :
D ⊆ E := by
simp_rw [← union]
exact subset_union_left | lemma | Topology | [
"Mathlib.Analysis.Normed.Module.RCLike.Real",
"Mathlib.Data.ENat.Basic",
"Mathlib.Logic.Equiv.PartialEquiv",
"Mathlib.Topology.MetricSpace.ProperSpace.Real"
] | Mathlib/Topology/CWComplex/Classical/Basic.lean | RelCWComplex.Subcomplex.base_subset | null |
@[simps! -isSimp, irreducible]
skeletonLT (C : Set X) {D : Set X} [RelCWComplex C D] (n : ℕ∞) : Subcomplex C :=
Subcomplex.mk' _ (D ∪ ⋃ (m : ℕ) (_ : m < n) (j : cell C m), closedCell m j)
(fun l ↦ {x : cell C l | l < n})
(by
intro l ⟨i, hi⟩
apply subset_union_of_subset_right
apply subset_i... | def | Topology | [
"Mathlib.Analysis.Normed.Module.RCLike.Real",
"Mathlib.Data.ENat.Basic",
"Mathlib.Logic.Equiv.PartialEquiv",
"Mathlib.Topology.MetricSpace.ProperSpace.Real"
] | Mathlib/Topology/CWComplex/Classical/Basic.lean | skeletonLT | A non-standard definition of the `n`-skeleton of a CW complex for `n ∈ ℕ ∪ {∞}`.
This allows the base case of induction to be about the base instead of being about the union of
the base and some points.
The standard `skeleton` is defined in terms of `skeletonLT`. `skeletonLT` is preferred
in statements. You should then... |
skeleton (C : Set X) {D : Set X} [RelCWComplex C D] (n : ℕ∞) : Subcomplex C :=
skeletonLT C (n + 1) | abbrev | Topology | [
"Mathlib.Analysis.Normed.Module.RCLike.Real",
"Mathlib.Data.ENat.Basic",
"Mathlib.Logic.Equiv.PartialEquiv",
"Mathlib.Topology.MetricSpace.ProperSpace.Real"
] | Mathlib/Topology/CWComplex/Classical/Basic.lean | skeleton | The `n`-skeleton of a CW complex, for `n ∈ ℕ ∪ {∞}`. For statements use `skeletonLT` instead
and then derive the statement about `skeleton`. |
RelCWComplex.skeletonLT_zero_eq_base [RelCWComplex C D] : skeletonLT C 0 = D := by
simp [coe_skeletonLT] | lemma | Topology | [
"Mathlib.Analysis.Normed.Module.RCLike.Real",
"Mathlib.Data.ENat.Basic",
"Mathlib.Logic.Equiv.PartialEquiv",
"Mathlib.Topology.MetricSpace.ProperSpace.Real"
] | Mathlib/Topology/CWComplex/Classical/Basic.lean | RelCWComplex.skeletonLT_zero_eq_base | null |
CWComplex.skeletonLT_zero_eq_empty [CWComplex C] : (skeletonLT C 0 : Set X) = ∅ :=
RelCWComplex.skeletonLT_zero_eq_base
@[simp] lemma RelCWComplex.skeletonLT_top [RelCWComplex C D] : skeletonLT C ⊤ = C := by
simp [coe_skeletonLT, union]
@[simp] lemma RelCWComplex.skeleton_top [RelCWComplex C D] : skeleton C ⊤ = C... | lemma | Topology | [
"Mathlib.Analysis.Normed.Module.RCLike.Real",
"Mathlib.Data.ENat.Basic",
"Mathlib.Logic.Equiv.PartialEquiv",
"Mathlib.Topology.MetricSpace.ProperSpace.Real"
] | Mathlib/Topology/CWComplex/Classical/Basic.lean | CWComplex.skeletonLT_zero_eq_empty | null |
RelCWComplex.skeletonLT_mono [RelCWComplex C D] {n m : ℕ∞} (h : m ≤ n) :
(skeletonLT C m : Set X) ⊆ skeletonLT C n := by
simp_rw [coe_skeletonLT]
apply union_subset_union_right
intro x xmem
simp_rw [mem_iUnion, exists_prop] at xmem ⊢
obtain ⟨l, lltm, xmeml⟩ := xmem
exact ⟨l, lt_of_lt_of_le lltm h, xmeml... | lemma | Topology | [
"Mathlib.Analysis.Normed.Module.RCLike.Real",
"Mathlib.Data.ENat.Basic",
"Mathlib.Logic.Equiv.PartialEquiv",
"Mathlib.Topology.MetricSpace.ProperSpace.Real"
] | Mathlib/Topology/CWComplex/Classical/Basic.lean | RelCWComplex.skeletonLT_mono | null |
RelCWComplex.skeletonLT_monotone [RelCWComplex C D] : Monotone (skeletonLT C) :=
fun _ _ h ↦ skeletonLT_mono h | lemma | Topology | [
"Mathlib.Analysis.Normed.Module.RCLike.Real",
"Mathlib.Data.ENat.Basic",
"Mathlib.Logic.Equiv.PartialEquiv",
"Mathlib.Topology.MetricSpace.ProperSpace.Real"
] | Mathlib/Topology/CWComplex/Classical/Basic.lean | RelCWComplex.skeletonLT_monotone | null |
RelCWComplex.skeleton_mono [RelCWComplex C D] {n m : ℕ∞} (h : m ≤ n) :
(skeleton C m : Set X) ⊆ skeleton C n :=
skeletonLT_mono (add_le_add_right h 1) | lemma | Topology | [
"Mathlib.Analysis.Normed.Module.RCLike.Real",
"Mathlib.Data.ENat.Basic",
"Mathlib.Logic.Equiv.PartialEquiv",
"Mathlib.Topology.MetricSpace.ProperSpace.Real"
] | Mathlib/Topology/CWComplex/Classical/Basic.lean | RelCWComplex.skeleton_mono | null |
RelCWComplex.skeleton_monotone [RelCWComplex C D] : Monotone (skeleton C) :=
fun _ _ h ↦ skeleton_mono h | lemma | Topology | [
"Mathlib.Analysis.Normed.Module.RCLike.Real",
"Mathlib.Data.ENat.Basic",
"Mathlib.Logic.Equiv.PartialEquiv",
"Mathlib.Topology.MetricSpace.ProperSpace.Real"
] | Mathlib/Topology/CWComplex/Classical/Basic.lean | RelCWComplex.skeleton_monotone | null |
RelCWComplex.closedCell_subset_skeletonLT [RelCWComplex C D] (n : ℕ) (j : cell C n) :
closedCell n j ⊆ skeletonLT C (n + 1) := by
intro x xmem
rw [coe_skeletonLT]
right
simp_rw [mem_iUnion, exists_prop]
refine ⟨n, (by norm_cast; exact lt_add_one n), ⟨j,xmem⟩⟩ | lemma | Topology | [
"Mathlib.Analysis.Normed.Module.RCLike.Real",
"Mathlib.Data.ENat.Basic",
"Mathlib.Logic.Equiv.PartialEquiv",
"Mathlib.Topology.MetricSpace.ProperSpace.Real"
] | Mathlib/Topology/CWComplex/Classical/Basic.lean | RelCWComplex.closedCell_subset_skeletonLT | null |
RelCWComplex.closedCell_subset_skeleton [RelCWComplex C D] (n : ℕ) (j : cell C n) :
closedCell n j ⊆ skeleton C n :=
closedCell_subset_skeletonLT n j | lemma | Topology | [
"Mathlib.Analysis.Normed.Module.RCLike.Real",
"Mathlib.Data.ENat.Basic",
"Mathlib.Logic.Equiv.PartialEquiv",
"Mathlib.Topology.MetricSpace.ProperSpace.Real"
] | Mathlib/Topology/CWComplex/Classical/Basic.lean | RelCWComplex.closedCell_subset_skeleton | null |
RelCWComplex.openCell_subset_skeletonLT [RelCWComplex C D] (n : ℕ) (j : cell C n) :
openCell n j ⊆ skeletonLT C (n + 1) :=
(openCell_subset_closedCell _ _).trans (closedCell_subset_skeletonLT _ _) | lemma | Topology | [
"Mathlib.Analysis.Normed.Module.RCLike.Real",
"Mathlib.Data.ENat.Basic",
"Mathlib.Logic.Equiv.PartialEquiv",
"Mathlib.Topology.MetricSpace.ProperSpace.Real"
] | Mathlib/Topology/CWComplex/Classical/Basic.lean | RelCWComplex.openCell_subset_skeletonLT | null |
RelCWComplex.openCell_subset_skeleton [RelCWComplex C D] (n : ℕ) (j : cell C n) :
openCell n j ⊆ skeleton C n :=
(openCell_subset_closedCell _ _).trans (closedCell_subset_skeleton _ _) | lemma | Topology | [
"Mathlib.Analysis.Normed.Module.RCLike.Real",
"Mathlib.Data.ENat.Basic",
"Mathlib.Logic.Equiv.PartialEquiv",
"Mathlib.Topology.MetricSpace.ProperSpace.Real"
] | Mathlib/Topology/CWComplex/Classical/Basic.lean | RelCWComplex.openCell_subset_skeleton | null |
RelCWComplex.cellFrontier_subset_skeletonLT [RelCWComplex C D] (n : ℕ) (j : cell C n) :
cellFrontier n j ⊆ skeletonLT C n := by
obtain ⟨I, hI⟩ := cellFrontier_subset_base_union_finite_closedCell n j
apply subset_trans hI
rw [coe_skeletonLT]
apply union_subset_union_right
intro x xmem
simp only [mem_iUni... | lemma | Topology | [
"Mathlib.Analysis.Normed.Module.RCLike.Real",
"Mathlib.Data.ENat.Basic",
"Mathlib.Logic.Equiv.PartialEquiv",
"Mathlib.Topology.MetricSpace.ProperSpace.Real"
] | Mathlib/Topology/CWComplex/Classical/Basic.lean | RelCWComplex.cellFrontier_subset_skeletonLT | null |
RelCWComplex.cellFrontier_subset_skeleton [RelCWComplex C D] (n : ℕ) (j : cell C (n + 1)) :
cellFrontier (n + 1) j ⊆ skeleton C n :=
cellFrontier_subset_skeletonLT _ _ | lemma | Topology | [
"Mathlib.Analysis.Normed.Module.RCLike.Real",
"Mathlib.Data.ENat.Basic",
"Mathlib.Logic.Equiv.PartialEquiv",
"Mathlib.Topology.MetricSpace.ProperSpace.Real"
] | Mathlib/Topology/CWComplex/Classical/Basic.lean | RelCWComplex.cellFrontier_subset_skeleton | null |
RelCWComplex.iUnion_cellFrontier_subset_skeletonLT [RelCWComplex C D] (l : ℕ) :
⋃ (j : cell C l), cellFrontier l j ⊆ skeletonLT C l :=
iUnion_subset (fun _ ↦ cellFrontier_subset_skeletonLT _ _) | lemma | Topology | [
"Mathlib.Analysis.Normed.Module.RCLike.Real",
"Mathlib.Data.ENat.Basic",
"Mathlib.Logic.Equiv.PartialEquiv",
"Mathlib.Topology.MetricSpace.ProperSpace.Real"
] | Mathlib/Topology/CWComplex/Classical/Basic.lean | RelCWComplex.iUnion_cellFrontier_subset_skeletonLT | null |
RelCWComplex.iUnion_cellFrontier_subset_skeleton [RelCWComplex C D] (l : ℕ) :
⋃ (j : cell C l), cellFrontier l j ⊆ skeleton C l :=
(iUnion_cellFrontier_subset_skeletonLT l).trans (skeletonLT_mono le_self_add) | lemma | Topology | [
"Mathlib.Analysis.Normed.Module.RCLike.Real",
"Mathlib.Data.ENat.Basic",
"Mathlib.Logic.Equiv.PartialEquiv",
"Mathlib.Topology.MetricSpace.ProperSpace.Real"
] | Mathlib/Topology/CWComplex/Classical/Basic.lean | RelCWComplex.iUnion_cellFrontier_subset_skeleton | null |
RelCWComplex.skeletonLT_union_iUnion_closedCell_eq_skeletonLT_succ [RelCWComplex C D]
(n : ℕ) :
(skeletonLT C n : Set X) ∪ ⋃ (j : cell C n), closedCell n j = skeletonLT C (n + 1) := by
rw [coe_skeletonLT, coe_skeletonLT, union_assoc]
congr
norm_cast
exact (biUnion_lt_succ _ _).symm | lemma | Topology | [
"Mathlib.Analysis.Normed.Module.RCLike.Real",
"Mathlib.Data.ENat.Basic",
"Mathlib.Logic.Equiv.PartialEquiv",
"Mathlib.Topology.MetricSpace.ProperSpace.Real"
] | Mathlib/Topology/CWComplex/Classical/Basic.lean | RelCWComplex.skeletonLT_union_iUnion_closedCell_eq_skeletonLT_succ | null |
RelCWComplex.skeleton_union_iUnion_closedCell_eq_skeleton_succ [RelCWComplex C D] (n : ℕ) :
(skeleton C n : Set X) ∪ ⋃ (j : cell C (n + 1)), closedCell (n + 1) j = skeleton C (n + 1) :=
skeletonLT_union_iUnion_closedCell_eq_skeletonLT_succ _ | lemma | Topology | [
"Mathlib.Analysis.Normed.Module.RCLike.Real",
"Mathlib.Data.ENat.Basic",
"Mathlib.Logic.Equiv.PartialEquiv",
"Mathlib.Topology.MetricSpace.ProperSpace.Real"
] | Mathlib/Topology/CWComplex/Classical/Basic.lean | RelCWComplex.skeleton_union_iUnion_closedCell_eq_skeleton_succ | null |
RelCWComplex.iUnion_openCell_eq_skeletonLT [RelCWComplex C D] (n : ℕ∞) :
D ∪ ⋃ (m : ℕ) (_ : m < n) (j : cell C m), openCell m j = skeletonLT C n :=
(coe_skeletonLT C _).symm ▸ RelCWComplex.iUnion_openCell_eq_iUnion_closedCell n | lemma | Topology | [
"Mathlib.Analysis.Normed.Module.RCLike.Real",
"Mathlib.Data.ENat.Basic",
"Mathlib.Logic.Equiv.PartialEquiv",
"Mathlib.Topology.MetricSpace.ProperSpace.Real"
] | Mathlib/Topology/CWComplex/Classical/Basic.lean | RelCWComplex.iUnion_openCell_eq_skeletonLT | A version of the definition of `skeletonLT` with open cells. |
CWComplex.iUnion_openCell_eq_skeletonLT [CWComplex C] (n : ℕ∞) :
⋃ (m : ℕ) (_ : m < n) (j : cell C m), openCell m j = skeletonLT C n := by
rw [← RelCWComplex.iUnion_openCell_eq_skeletonLT, empty_union] | lemma | Topology | [
"Mathlib.Analysis.Normed.Module.RCLike.Real",
"Mathlib.Data.ENat.Basic",
"Mathlib.Logic.Equiv.PartialEquiv",
"Mathlib.Topology.MetricSpace.ProperSpace.Real"
] | Mathlib/Topology/CWComplex/Classical/Basic.lean | CWComplex.iUnion_openCell_eq_skeletonLT | null |
RelCWComplex.iUnion_openCell_eq_skeleton [RelCWComplex C D] (n : ℕ∞) :
D ∪ ⋃ (m : ℕ) (_ : m < n + 1) (j : cell C m), openCell m j = skeleton C n :=
iUnion_openCell_eq_skeletonLT _ | lemma | Topology | [
"Mathlib.Analysis.Normed.Module.RCLike.Real",
"Mathlib.Data.ENat.Basic",
"Mathlib.Logic.Equiv.PartialEquiv",
"Mathlib.Topology.MetricSpace.ProperSpace.Real"
] | Mathlib/Topology/CWComplex/Classical/Basic.lean | RelCWComplex.iUnion_openCell_eq_skeleton | null |
CWComplex.iUnion_openCell_eq_skeleton [CWComplex C] (n : ℕ∞) :
⋃ (m : ℕ) (_ : m < n + 1) (j : cell C m), openCell m j = skeleton C n :=
iUnion_openCell_eq_skeletonLT _ | lemma | Topology | [
"Mathlib.Analysis.Normed.Module.RCLike.Real",
"Mathlib.Data.ENat.Basic",
"Mathlib.Logic.Equiv.PartialEquiv",
"Mathlib.Topology.MetricSpace.ProperSpace.Real"
] | Mathlib/Topology/CWComplex/Classical/Basic.lean | CWComplex.iUnion_openCell_eq_skeleton | null |
RelCWComplex.iUnion_skeletonLT_eq_complex [RelCWComplex C D] :
⋃ (n : ℕ), skeletonLT C n = C := by
apply subset_antisymm (iUnion_subset_iff.2 fun _ ↦ (skeletonLT C _).subset_complex)
simp_rw [← union_iUnion_openCell_eq_complex, union_subset_iff, iUnion₂_subset_iff]
exact ⟨subset_iUnion_of_subset 0 (skeletonLT... | lemma | Topology | [
"Mathlib.Analysis.Normed.Module.RCLike.Real",
"Mathlib.Data.ENat.Basic",
"Mathlib.Logic.Equiv.PartialEquiv",
"Mathlib.Topology.MetricSpace.ProperSpace.Real"
] | Mathlib/Topology/CWComplex/Classical/Basic.lean | RelCWComplex.iUnion_skeletonLT_eq_complex | null |
RelCWComplex.iUnion_skeleton_eq_complex [RelCWComplex C D] :
⋃ (n : ℕ), skeleton C n = C := by
apply subset_antisymm (iUnion_subset_iff.2 fun _ ↦ (skeleton C _).subset_complex)
simp_rw [← union_iUnion_openCell_eq_complex, union_subset_iff, iUnion₂_subset_iff]
exact ⟨subset_iUnion_of_subset 0 (skeleton C 0).ba... | lemma | Topology | [
"Mathlib.Analysis.Normed.Module.RCLike.Real",
"Mathlib.Data.ENat.Basic",
"Mathlib.Logic.Equiv.PartialEquiv",
"Mathlib.Topology.MetricSpace.ProperSpace.Real"
] | Mathlib/Topology/CWComplex/Classical/Basic.lean | RelCWComplex.iUnion_skeleton_eq_complex | null |
RelCWComplex.mem_skeletonLT_iff [RelCWComplex C D] {n : ℕ∞} {x : X} :
x ∈ skeletonLT C n ↔ x ∈ D ∨ ∃ (m : ℕ) (_ : m < n) (j : cell C m), x ∈ openCell m j := by
simp [← SetLike.mem_coe, ← iUnion_openCell_eq_skeletonLT] | lemma | Topology | [
"Mathlib.Analysis.Normed.Module.RCLike.Real",
"Mathlib.Data.ENat.Basic",
"Mathlib.Logic.Equiv.PartialEquiv",
"Mathlib.Topology.MetricSpace.ProperSpace.Real"
] | Mathlib/Topology/CWComplex/Classical/Basic.lean | RelCWComplex.mem_skeletonLT_iff | null |
CWComplex.mem_skeletonLT_iff [CWComplex C] {n : ℕ∞} {x : X} :
x ∈ skeletonLT C n ↔ ∃ (m : ℕ) (_ : m < n) (j : cell C m), x ∈ openCell m j := by
simp [← SetLike.mem_coe, ← iUnion_openCell_eq_skeletonLT] | lemma | Topology | [
"Mathlib.Analysis.Normed.Module.RCLike.Real",
"Mathlib.Data.ENat.Basic",
"Mathlib.Logic.Equiv.PartialEquiv",
"Mathlib.Topology.MetricSpace.ProperSpace.Real"
] | Mathlib/Topology/CWComplex/Classical/Basic.lean | CWComplex.mem_skeletonLT_iff | null |
RelCWComplex.mem_skeleton_iff [RelCWComplex C D] {n : ℕ∞} {x : X} :
x ∈ skeleton C n ↔ x ∈ D ∨ ∃ (m : ℕ) (_ : m ≤ n) (j : cell C m), x ∈ openCell m j := by
rw [skeleton, mem_skeletonLT_iff]
suffices ∀ (m : ℕ), m < n + 1 ↔ m ≤ n by simp_rw [this]
intro m
cases n
· simp
· rw [← Nat.cast_one, ← Nat.cast_ad... | lemma | Topology | [
"Mathlib.Analysis.Normed.Module.RCLike.Real",
"Mathlib.Data.ENat.Basic",
"Mathlib.Logic.Equiv.PartialEquiv",
"Mathlib.Topology.MetricSpace.ProperSpace.Real"
] | Mathlib/Topology/CWComplex/Classical/Basic.lean | RelCWComplex.mem_skeleton_iff | null |
CWComplex.exists_mem_openCell_of_mem_skeleton [CWComplex C] {n : ℕ∞} {x : X} :
x ∈ skeleton C n ↔ ∃ (m : ℕ) (_ : m ≤ n) (j : cell C m), x ∈ openCell m j := by
rw [RelCWComplex.mem_skeleton_iff, mem_empty_iff_false, false_or] | lemma | Topology | [
"Mathlib.Analysis.Normed.Module.RCLike.Real",
"Mathlib.Data.ENat.Basic",
"Mathlib.Logic.Equiv.PartialEquiv",
"Mathlib.Topology.MetricSpace.ProperSpace.Real"
] | Mathlib/Topology/CWComplex/Classical/Basic.lean | CWComplex.exists_mem_openCell_of_mem_skeleton | null |
RelCWComplex.disjoint_skeletonLT_openCell [RelCWComplex C D] {n : ℕ∞} {m : ℕ}
{j : cell C m} (hnm : n ≤ m) : Disjoint (skeletonLT C n : Set X) (openCell m j) := by
simp_rw [← iUnion_openCell_eq_skeletonLT, disjoint_union_left, disjoint_iUnion_left]
refine ⟨(disjointBase m j).symm, ?_⟩
intro l hln i
apply di... | lemma | Topology | [
"Mathlib.Analysis.Normed.Module.RCLike.Real",
"Mathlib.Data.ENat.Basic",
"Mathlib.Logic.Equiv.PartialEquiv",
"Mathlib.Topology.MetricSpace.ProperSpace.Real"
] | Mathlib/Topology/CWComplex/Classical/Basic.lean | RelCWComplex.disjoint_skeletonLT_openCell | A skeleton and an open cell of a higher dimension are disjoint. |
RelCWComplex.disjoint_skeleton_openCell [RelCWComplex C D] {n : ℕ∞} {m : ℕ}
{j : cell C m} (nlem : n < m) : Disjoint (skeleton C n : Set X) (openCell m j) :=
disjoint_skeletonLT_openCell (Order.add_one_le_of_lt nlem) | lemma | Topology | [
"Mathlib.Analysis.Normed.Module.RCLike.Real",
"Mathlib.Data.ENat.Basic",
"Mathlib.Logic.Equiv.PartialEquiv",
"Mathlib.Topology.MetricSpace.ProperSpace.Real"
] | Mathlib/Topology/CWComplex/Classical/Basic.lean | RelCWComplex.disjoint_skeleton_openCell | A skeleton and an open cell of a higher dimension are disjoint. |
RelCWComplex.skeletonLT_inter_closedCell_eq_skeletonLT_inter_cellFrontier [RelCWComplex C D]
{n : ℕ∞} {m : ℕ} {j : cell C m} (hnm : n ≤ m) :
(skeletonLT C n : Set X) ∩ closedCell m j = (skeletonLT C n : Set X) ∩ cellFrontier m j := by
refine subset_antisymm ?_ (inter_subset_inter_right _ (cellFrontier_subset_... | lemma | Topology | [
"Mathlib.Analysis.Normed.Module.RCLike.Real",
"Mathlib.Data.ENat.Basic",
"Mathlib.Logic.Equiv.PartialEquiv",
"Mathlib.Topology.MetricSpace.ProperSpace.Real"
] | Mathlib/Topology/CWComplex/Classical/Basic.lean | RelCWComplex.skeletonLT_inter_closedCell_eq_skeletonLT_inter_cellFrontier | A skeleton intersected with a closed cell of a higher dimension is the skeleton intersected with
the boundary of the cell. |
RelCWComplex.skeleton_inter_closedCell_eq_skeleton_inter_cellFrontier [RelCWComplex C D]
{n : ℕ∞} {m : ℕ} {j : cell C m} (hnm : n < m) :
(skeleton C n : Set X) ∩ closedCell m j = (skeleton C n : Set X) ∩ cellFrontier m j :=
skeletonLT_inter_closedCell_eq_skeletonLT_inter_cellFrontier (Order.add_one_le_of_lt h... | lemma | Topology | [
"Mathlib.Analysis.Normed.Module.RCLike.Real",
"Mathlib.Data.ENat.Basic",
"Mathlib.Logic.Equiv.PartialEquiv",
"Mathlib.Topology.MetricSpace.ProperSpace.Real"
] | Mathlib/Topology/CWComplex/Classical/Basic.lean | RelCWComplex.skeleton_inter_closedCell_eq_skeleton_inter_cellFrontier | Version of `skeletonLT_inter_closedCell_eq_skeletonLT_inter_cellFrontier` using `skeleton`. |
RelCWComplex.disjoint_interior_base_closedCell [T2Space X] [RelCWComplex C D] {n : ℕ}
{j : cell C n} : Disjoint (interior D) (closedCell n j) := by
rw [disjoint_iff_inter_eq_empty]
by_contra h
push_neg at h
rw [← closure_openCell_eq_closedCell, inter_comm,
closure_inter_open_nonempty_iff isOpen_interior... | lemma | Topology | [
"Mathlib.Analysis.Normed.Module.RCLike.Real",
"Mathlib.Data.ENat.Basic",
"Mathlib.Logic.Equiv.PartialEquiv",
"Mathlib.Topology.MetricSpace.ProperSpace.Real"
] | Mathlib/Topology/CWComplex/Classical/Basic.lean | RelCWComplex.disjoint_interior_base_closedCell | null |
RelCWComplex.disjoint_interior_base_iUnion_closedCell [T2Space X] [RelCWComplex C D] :
Disjoint (interior D) (⋃ (n : ℕ) (j : cell C n), closedCell n j) := by
simp_rw [disjoint_iff_inter_eq_empty, inter_iUnion, disjoint_interior_base_closedCell.inter_eq,
iUnion_empty] | lemma | Topology | [
"Mathlib.Analysis.Normed.Module.RCLike.Real",
"Mathlib.Data.ENat.Basic",
"Mathlib.Logic.Equiv.PartialEquiv",
"Mathlib.Topology.MetricSpace.ProperSpace.Real"
] | Mathlib/Topology/CWComplex/Classical/Basic.lean | RelCWComplex.disjoint_interior_base_iUnion_closedCell | null |
RelCWComplex.FiniteDimensional.{u} {X : Type u} [TopologicalSpace X] (C : Set X) {D : Set X}
[RelCWComplex C D] : Prop where
/-- For some natural number `n`, the type `cell C m` is empty for all `m ≥ n`. -/
eventually_isEmpty_cell : ∀ᶠ n in Filter.atTop, IsEmpty (cell C n) | class | Topology | [
"Mathlib.Topology.CWComplex.Classical.Basic"
] | Mathlib/Topology/CWComplex/Classical/Finite.lean | RelCWComplex.FiniteDimensional. | A CW complex is finite dimensional if `cell C n` is empty for all but finitely many `n`. |
RelCWComplex.FiniteType.{u} {X : Type u} [TopologicalSpace X] (C : Set X) {D : Set X}
[RelCWComplex C D] : Prop where
/-- `cell C n` is finite for every `n`. -/
finite_cell (n : ℕ) : Finite (cell C n) | class | Topology | [
"Mathlib.Topology.CWComplex.Classical.Basic"
] | Mathlib/Topology/CWComplex/Classical/Finite.lean | RelCWComplex.FiniteType. | A CW complex is of finite type if `cell C n` is finite for every `n`. |
RelCWComplex.Finite {X : Type*} [TopologicalSpace X] (C : Set X) {D : Set X}
[RelCWComplex C D] extends FiniteDimensional C, FiniteType C
variable {X : Type*} [TopologicalSpace X] (C : Set X) {D : Set X} [RelCWComplex C D] | class | Topology | [
"Mathlib.Topology.CWComplex.Classical.Basic"
] | Mathlib/Topology/CWComplex/Classical/Finite.lean | RelCWComplex.Finite | A CW complex is finite if it is finite dimensional and of finite type. |
RelCWComplex.finite_of_finiteDimensional_finiteType [FiniteDimensional C]
[FiniteType C] : Finite C where
eventually_isEmpty_cell := FiniteDimensional.eventually_isEmpty_cell
finite_cell n := FiniteType.finite_cell n | lemma | Topology | [
"Mathlib.Topology.CWComplex.Classical.Basic"
] | Mathlib/Topology/CWComplex/Classical/Finite.lean | RelCWComplex.finite_of_finiteDimensional_finiteType | null |
@[simps -isSimp]
RelCWComplex.mkFiniteType.{u} {X : Type u} [TopologicalSpace X] (C : Set X)
(D : outParam (Set X))
(cell : (n : ℕ) → Type u) (map : (n : ℕ) → (i : cell n) → PartialEquiv (Fin n → ℝ) X)
(finite_cell : ∀ (n : ℕ), _root_.Finite (cell n))
(source_eq : ∀ (n : ℕ) (i : cell n), (map n i).sourc... | def | Topology | [
"Mathlib.Topology.CWComplex.Classical.Basic"
] | Mathlib/Topology/CWComplex/Classical/Finite.lean | RelCWComplex.mkFiniteType. | If we want to construct a relative CW complex of finite type, we can add the condition
`finite_cell` and relax the condition `mapsTo`. |
RelCWComplex.finiteType_mkFiniteType.{u} {X : Type u} [TopologicalSpace X] (C : Set X)
(D : outParam (Set X))
(cell : (n : ℕ) → Type u) (map : (n : ℕ) → (i : cell n) → PartialEquiv (Fin n → ℝ) X)
(finite_cell : ∀ (n : ℕ), _root_.Finite (cell n))
(source_eq : ∀ (n : ℕ) (i : cell n), (map n i).source = ba... | lemma | Topology | [
"Mathlib.Topology.CWComplex.Classical.Basic"
] | Mathlib/Topology/CWComplex/Classical/Finite.lean | RelCWComplex.finiteType_mkFiniteType. | A CW complex that was constructed using `RelCWComplex.mkFiniteType` is of finite type. |
@[simps -isSimp]
CWComplex.mkFiniteType.{u} {X : Type u} [TopologicalSpace X] (C : Set X)
(cell : (n : ℕ) → Type u) (map : (n : ℕ) → (i : cell n) → PartialEquiv (Fin n → ℝ) X)
(finite_cell : ∀ (n : ℕ), _root_.Finite (cell n))
(source_eq : ∀ (n : ℕ) (i : cell n), (map n i).source = ball 0 1)
(continuousO... | def | Topology | [
"Mathlib.Topology.CWComplex.Classical.Basic"
] | Mathlib/Topology/CWComplex/Classical/Finite.lean | CWComplex.mkFiniteType. | If we want to construct a CW complex of finite type, we can add the condition `finite_cell` and
relax the condition `mapsTo`. |
CWComplex.finiteType_mkFiniteType.{u} {X : Type u} [TopologicalSpace X] (C : Set X)
(cell : (n : ℕ) → Type u) (map : (n : ℕ) → (i : cell n) → PartialEquiv (Fin n → ℝ) X)
(finite_cell : ∀ (n : ℕ), _root_.Finite (cell n))
(source_eq : ∀ (n : ℕ) (i : cell n), (map n i).source = ball 0 1)
(continuousOn : ∀ ... | lemma | Topology | [
"Mathlib.Topology.CWComplex.Classical.Basic"
] | Mathlib/Topology/CWComplex/Classical/Finite.lean | CWComplex.finiteType_mkFiniteType. | A CW complex that was constructed using `CWComplex.mkFiniteType` is of finite type. |
@[simps -isSimp]
RelCWComplex.mkFinite.{u} {X : Type u} [TopologicalSpace X] (C : Set X)
(D : outParam (Set X)) (cell : (n : ℕ) → Type u)
(map : (n : ℕ) → (i : cell n) → PartialEquiv (Fin n → ℝ) X)
(eventually_isEmpty_cell : ∀ᶠ n in Filter.atTop, IsEmpty (cell n))
(finite_cell : ∀ (n : ℕ), _root_.Finite... | def | Topology | [
"Mathlib.Topology.CWComplex.Classical.Basic"
] | Mathlib/Topology/CWComplex/Classical/Finite.lean | RelCWComplex.mkFinite. | If we want to construct a finite relative CW complex we can add the conditions
`eventually_isEmpty_cell` and `finite_cell`, relax the condition `mapsTo` and remove the condition
`closed'`. |
RelCWComplex.finite_mkFinite.{u} {X : Type u} [TopologicalSpace X] (C : Set X)
(D : outParam (Set X)) (cell : (n : ℕ) → Type u)
(map : (n : ℕ) → (i : cell n) → PartialEquiv (Fin n → ℝ) X)
(eventually_isEmpty_cell : ∀ᶠ n in Filter.atTop, IsEmpty (cell n))
(finite_cell : ∀ (n : ℕ), _root_.Finite (cell n))... | lemma | Topology | [
"Mathlib.Topology.CWComplex.Classical.Basic"
] | Mathlib/Topology/CWComplex/Classical/Finite.lean | RelCWComplex.finite_mkFinite. | A CW complex that was constructed using `RelCWComplex.mkFinite` is finite. |
@[simps! -isSimp]
CWComplex.mkFinite.{u} {X : Type u} [TopologicalSpace X] (C : Set X)
(cell : (n : ℕ) → Type u) (map : (n : ℕ) → (i : cell n) → PartialEquiv (Fin n → ℝ) X)
(eventually_isEmpty_cell : ∀ᶠ n in Filter.atTop, IsEmpty (cell n))
(finite_cell : ∀ (n : ℕ), _root_.Finite (cell n))
(source_eq : ∀... | def | Topology | [
"Mathlib.Topology.CWComplex.Classical.Basic"
] | Mathlib/Topology/CWComplex/Classical/Finite.lean | CWComplex.mkFinite. | If we want to construct a finite CW complex we can add the conditions `eventually_isEmpty_cell`
and `finite_cell`, relax the condition `mapsTo` and remove the condition `closed'`. |
CWComplex.finite_mkFinite.{u} {X : Type u} [TopologicalSpace X] (C : Set X)
(cell : (n : ℕ) → Type u) (map : (n : ℕ) → (i : cell n) → PartialEquiv (Fin n → ℝ) X)
(eventually_isEmpty_cell : ∀ᶠ n in Filter.atTop, IsEmpty (cell n))
(finite_cell : ∀ (n : ℕ), _root_.Finite (cell n))
(source_eq : ∀ (n : ℕ) (i... | lemma | Topology | [
"Mathlib.Topology.CWComplex.Classical.Basic"
] | Mathlib/Topology/CWComplex/Classical/Finite.lean | CWComplex.finite_mkFinite. | A CW complex that was constructed using `CWComplex.mkFinite` is finite. |
RelCWComplex.finite_of_finite_cells (finite : _root_.Finite (Σ n, cell C n)) : Finite C where
eventually_isEmpty_cell := by
simp only [Filter.eventually_atTop, ge_iff_le]
by_cases h : IsEmpty (Σ n, cell C n)
· exact ⟨0, by simp_all⟩
push_neg at h
have _ := Fintype.ofFinite (Σ n, cell C n)
clas... | lemma | Topology | [
"Mathlib.Topology.CWComplex.Classical.Basic"
] | Mathlib/Topology/CWComplex/Classical/Finite.lean | RelCWComplex.finite_of_finite_cells | If the collection of all cells (of any dimension) of a relative CW complex `C` is finite, then
`C` is finite as a CW complex. |
RelCWComplex.finite_cells_of_finite [finite : Finite C] : _root_.Finite (Σ n, cell C n) := by
have h := finite.eventually_isEmpty_cell
have _ := finite.finite_cell
simp only [Filter.eventually_atTop, ge_iff_le] at h
rcases h with ⟨n, hn⟩
have (m) (j : cell C m) : m < n := by
by_contra h
exact (hn m (n... | lemma | Topology | [
"Mathlib.Topology.CWComplex.Classical.Basic"
] | Mathlib/Topology/CWComplex/Classical/Finite.lean | RelCWComplex.finite_cells_of_finite | If `C` is finite as a CW complex then the collection of all cells (of any dimension) is
finite. |
RelCWComplex.finite_iff_finite_cells : Finite C ↔ _root_.Finite (Σ n, cell C n) :=
⟨fun h ↦ finite_cells_of_finite (finite := h), finite_of_finite_cells⟩ | lemma | Topology | [
"Mathlib.Topology.CWComplex.Classical.Basic"
] | Mathlib/Topology/CWComplex/Classical/Finite.lean | RelCWComplex.finite_iff_finite_cells | A CW complex is finite iff the total number of its cells is finite. |
RelCWComplex.Subcomplex.closedCell_subset_of_mem [T2Space X] [RelCWComplex C D]
(E : Subcomplex C) {n : ℕ} {i : cell C n} (hi : i ∈ E.I n) :
closedCell n i ⊆ E := by
rw [← closure_openCell_eq_closedCell, E.closed.closure_subset_iff, ← E.union]
apply subset_union_of_subset_right
exact subset_iUnion_of_subs... | lemma | Topology | [
"Mathlib.Topology.CWComplex.Classical.Finite",
"Mathlib.Analysis.Normed.Module.RCLike.Real"
] | Mathlib/Topology/CWComplex/Classical/Subcomplex.lean | RelCWComplex.Subcomplex.closedCell_subset_of_mem | null |
RelCWComplex.Subcomplex.openCell_subset_of_mem [T2Space X] [RelCWComplex C D]
(E : Subcomplex C) {n : ℕ} {i : cell C n} (hi : i ∈ E.I n) :
openCell n i ⊆ E :=
(openCell_subset_closedCell n i).trans (closedCell_subset_of_mem E hi) | lemma | Topology | [
"Mathlib.Topology.CWComplex.Classical.Finite",
"Mathlib.Analysis.Normed.Module.RCLike.Real"
] | Mathlib/Topology/CWComplex/Classical/Subcomplex.lean | RelCWComplex.Subcomplex.openCell_subset_of_mem | null |
RelCWComplex.Subcomplex.cellFrontier_subset_of_mem [T2Space X] [RelCWComplex C D]
(E : Subcomplex C) {n : ℕ} {i : cell C n} (hi : i ∈ E.I n) :
cellFrontier n i ⊆ E :=
(cellFrontier_subset_closedCell n i).trans (closedCell_subset_of_mem E hi) | lemma | Topology | [
"Mathlib.Topology.CWComplex.Classical.Finite",
"Mathlib.Analysis.Normed.Module.RCLike.Real"
] | Mathlib/Topology/CWComplex/Classical/Subcomplex.lean | RelCWComplex.Subcomplex.cellFrontier_subset_of_mem | null |
RelCWComplex.Subcomplex.union_closedCell [T2Space X] [RelCWComplex C D] (E : Subcomplex C) :
D ∪ ⋃ (n : ℕ) (j : E.I n), closedCell (C := C) n j = E := by
apply subset_antisymm
· apply union_subset E.base_subset
exact iUnion₂_subset fun n i ↦ closedCell_subset_of_mem E i.2
· rw [← E.union]
apply union_... | lemma | Topology | [
"Mathlib.Topology.CWComplex.Classical.Finite",
"Mathlib.Analysis.Normed.Module.RCLike.Real"
] | Mathlib/Topology/CWComplex/Classical/Subcomplex.lean | RelCWComplex.Subcomplex.union_closedCell | A subcomplex is the union of its closed cells and its base. |
CWComplex.Subcomplex.union_closedCell [T2Space X] [CWComplex C] (E : Subcomplex C) :
⋃ (n : ℕ) (j : E.I n), closedCell (C := C) n j = E :=
(empty_union _ ).symm.trans (RelCWComplex.Subcomplex.union_closedCell E) | lemma | Topology | [
"Mathlib.Topology.CWComplex.Classical.Finite",
"Mathlib.Analysis.Normed.Module.RCLike.Real"
] | Mathlib/Topology/CWComplex/Classical/Subcomplex.lean | CWComplex.Subcomplex.union_closedCell | A subcomplex is the union of its closed cells. |
RelCWComplex.Subcomplex.disjoint_openCell_subcomplex_of_not_mem [RelCWComplex C D]
(E : Subcomplex C) {n : ℕ} {i : cell C n} (h : i ∉ E.I n) : Disjoint (openCell n i) E := by
simp_rw [← union, disjoint_union_right, disjoint_iUnion_right]
exact ⟨disjointBase n i , fun _ _ ↦ disjoint_openCell_of_ne (by aesop)⟩
op... | lemma | Topology | [
"Mathlib.Topology.CWComplex.Classical.Finite",
"Mathlib.Analysis.Normed.Module.RCLike.Real"
] | Mathlib/Topology/CWComplex/Classical/Subcomplex.lean | RelCWComplex.Subcomplex.disjoint_openCell_subcomplex_of_not_mem | null |
@[simps]
RelCWComplex.Subcomplex.instRelCWComplex [T2Space X] [RelCWComplex C D]
(E : Subcomplex C) : RelCWComplex E D where
cell n := E.I n
map n i := map (C := C) n i
source_eq n i := source_eq (C := C) n i
continuousOn n i := continuousOn (C := C) n i
continuousOn_symm n i := continuousOn_symm (C := C)... | instance | Topology | [
"Mathlib.Topology.CWComplex.Classical.Finite",
"Mathlib.Analysis.Normed.Module.RCLike.Real"
] | Mathlib/Topology/CWComplex/Classical/Subcomplex.lean | RelCWComplex.Subcomplex.instRelCWComplex | A subcomplex is again a CW complex. |
CWComplex.Subcomplex.instCWComplex [T2Space X] [CWComplex C] (E : Subcomplex C) :
CWComplex (E : Set X) :=
RelCWComplex.toCWComplex (E : Set X)
@[simp] | instance | Topology | [
"Mathlib.Topology.CWComplex.Classical.Finite",
"Mathlib.Analysis.Normed.Module.RCLike.Real"
] | Mathlib/Topology/CWComplex/Classical/Subcomplex.lean | CWComplex.Subcomplex.instCWComplex | A subcomplex is again a CW complex. |
CWComplex.Subcomplex.cell_def [T2Space X] [CWComplex C] (E : Subcomplex C)
(n : ℕ) : cell (E : Set X) n = E.I (C := C) n :=
rfl
@[simp] | lemma | Topology | [
"Mathlib.Topology.CWComplex.Classical.Finite",
"Mathlib.Analysis.Normed.Module.RCLike.Real"
] | Mathlib/Topology/CWComplex/Classical/Subcomplex.lean | CWComplex.Subcomplex.cell_def | null |
CWComplex.Subcomplex.map_def [T2Space X] [CWComplex C] (E : Subcomplex C) (n : ℕ)
(i : E.I n) : map (C := E) n i = map (C := C) n i :=
rfl
@[simp] | lemma | Topology | [
"Mathlib.Topology.CWComplex.Classical.Finite",
"Mathlib.Analysis.Normed.Module.RCLike.Real"
] | Mathlib/Topology/CWComplex/Classical/Subcomplex.lean | CWComplex.Subcomplex.map_def | null |
RelCWComplex.Subcomplex.openCell_eq [T2Space X] [RelCWComplex C D] (E : Subcomplex C) (n : ℕ)
(i : E.I n) : openCell (C := E) n i = openCell n (i : cell C n) := by
rfl
@[simp] | lemma | Topology | [
"Mathlib.Topology.CWComplex.Classical.Finite",
"Mathlib.Analysis.Normed.Module.RCLike.Real"
] | Mathlib/Topology/CWComplex/Classical/Subcomplex.lean | RelCWComplex.Subcomplex.openCell_eq | null |
RelCWComplex.Subcomplex.closedCell_eq [T2Space X] [RelCWComplex C D] (E : Subcomplex C)
(n : ℕ) (i : E.I n) : closedCell (C := E) n i = closedCell n (i : cell C n) := by
rfl
@[simp] | lemma | Topology | [
"Mathlib.Topology.CWComplex.Classical.Finite",
"Mathlib.Analysis.Normed.Module.RCLike.Real"
] | Mathlib/Topology/CWComplex/Classical/Subcomplex.lean | RelCWComplex.Subcomplex.closedCell_eq | null |
RelCWComplex.Subcomplex.cellFrontier_eq [T2Space X] [RelCWComplex C D] (E : Subcomplex C)
(n : ℕ) (i : E.I n) : cellFrontier (C := E) n i = cellFrontier n (i : cell C n) := by
rfl | lemma | Topology | [
"Mathlib.Topology.CWComplex.Classical.Finite",
"Mathlib.Analysis.Normed.Module.RCLike.Real"
] | Mathlib/Topology/CWComplex/Classical/Subcomplex.lean | RelCWComplex.Subcomplex.cellFrontier_eq | null |
RelCWComplex.Subcomplex.finiteType_subcomplex_of_finiteType [T2Space X]
[RelCWComplex C D] [FiniteType C] (E : Subcomplex C) : FiniteType (E : Set X) where
finite_cell n :=
let _ := FiniteType.finite_cell (C := C) (D := D) n
Subtype.finite | instance | Topology | [
"Mathlib.Topology.CWComplex.Classical.Finite",
"Mathlib.Analysis.Normed.Module.RCLike.Real"
] | Mathlib/Topology/CWComplex/Classical/Subcomplex.lean | RelCWComplex.Subcomplex.finiteType_subcomplex_of_finiteType | null |
RelCWComplex.Subcomplex.finiteDimensional_subcomplex_of_finiteDimensional
[T2Space X] [RelCWComplex C D] [FiniteDimensional C] (E : Subcomplex C) :
FiniteDimensional (E : Set X) where
eventually_isEmpty_cell := by
filter_upwards [FiniteDimensional.eventually_isEmpty_cell (C := C) (D := D)] with n hn
s... | instance | Topology | [
"Mathlib.Topology.CWComplex.Classical.Finite",
"Mathlib.Analysis.Normed.Module.RCLike.Real"
] | Mathlib/Topology/CWComplex/Classical/Subcomplex.lean | RelCWComplex.Subcomplex.finiteDimensional_subcomplex_of_finiteDimensional | null |
RelCWComplex.Subcomplex.finite_subcomplex_of_finite [T2Space X] [RelCWComplex C D]
[Finite C] (E : Subcomplex C) : Finite (E : Set X) :=
finite_of_finiteDimensional_finiteType _ | instance | Topology | [
"Mathlib.Topology.CWComplex.Classical.Finite",
"Mathlib.Analysis.Normed.Module.RCLike.Real"
] | Mathlib/Topology/CWComplex/Classical/Subcomplex.lean | RelCWComplex.Subcomplex.finite_subcomplex_of_finite | A subcomplex of a finite CW complex is again finite. |
continuous_right_toIcoMod : ContinuousWithinAt (toIcoMod hp a) (Ici x) x := by
intro s h
rw [Filter.mem_map, mem_nhdsWithin_iff_exists_mem_nhds_inter]
haveI : Nontrivial 𝕜 := ⟨⟨0, p, hp.ne⟩⟩
simp_rw [mem_nhds_iff_exists_Ioo_subset] at h ⊢
obtain ⟨l, u, hxI, hIs⟩ := h
let d := toIcoDiv hp a x • p
have hd ... | theorem | Topology | [
"Mathlib.Algebra.Order.ToIntervalMod",
"Mathlib.Algebra.Ring.AddAut",
"Mathlib.Data.Nat.Totient",
"Mathlib.GroupTheory.Divisible",
"Mathlib.Topology.Algebra.IsUniformGroup.Basic",
"Mathlib.Topology.Algebra.Order.Field",
"Mathlib.Topology.IsLocalHomeomorph",
"Mathlib.Topology.Order.T5"
] | Mathlib/Topology/Instances/AddCircle/Defs.lean | continuous_right_toIcoMod | null |
continuous_left_toIocMod : ContinuousWithinAt (toIocMod hp a) (Iic x) x := by
rw [(funext fun y => Eq.trans (by rw [neg_neg]) <| toIocMod_neg _ _ _ :
toIocMod hp a = (fun x => p - x) ∘ toIcoMod hp (-a) ∘ Neg.neg)]
exact
(continuous_sub_left _).continuousAt.comp_continuousWithinAt <|
(continuous_righ... | theorem | Topology | [
"Mathlib.Algebra.Order.ToIntervalMod",
"Mathlib.Algebra.Ring.AddAut",
"Mathlib.Data.Nat.Totient",
"Mathlib.GroupTheory.Divisible",
"Mathlib.Topology.Algebra.IsUniformGroup.Basic",
"Mathlib.Topology.Algebra.Order.Field",
"Mathlib.Topology.IsLocalHomeomorph",
"Mathlib.Topology.Order.T5"
] | Mathlib/Topology/Instances/AddCircle/Defs.lean | continuous_left_toIocMod | null |
toIcoMod_eventuallyEq_toIocMod (hx : (x : 𝕜 ⧸ zmultiples p) ≠ a) :
toIcoMod hp a =ᶠ[𝓝 x] toIocMod hp a :=
IsOpen.mem_nhds
(by
rw [Ico_eq_locus_Ioc_eq_iUnion_Ioo]
exact isOpen_iUnion fun i => isOpen_Ioo) <|
(not_modEq_iff_toIcoMod_eq_toIocMod hp).1 <| not_modEq_iff_ne_mod_zmultiples.2 h... | theorem | Topology | [
"Mathlib.Algebra.Order.ToIntervalMod",
"Mathlib.Algebra.Ring.AddAut",
"Mathlib.Data.Nat.Totient",
"Mathlib.GroupTheory.Divisible",
"Mathlib.Topology.Algebra.IsUniformGroup.Basic",
"Mathlib.Topology.Algebra.Order.Field",
"Mathlib.Topology.IsLocalHomeomorph",
"Mathlib.Topology.Order.T5"
] | Mathlib/Topology/Instances/AddCircle/Defs.lean | toIcoMod_eventuallyEq_toIocMod | null |
continuousAt_toIcoMod (hx : (x : 𝕜 ⧸ zmultiples p) ≠ a) : ContinuousAt (toIcoMod hp a) x :=
let h := toIcoMod_eventuallyEq_toIocMod hp a hx
continuousAt_iff_continuous_left_right.2 <|
⟨(continuous_left_toIocMod hp a x).congr_of_eventuallyEq (h.filter_mono nhdsWithin_le_nhds)
h.eq_of_nhds,
continu... | theorem | Topology | [
"Mathlib.Algebra.Order.ToIntervalMod",
"Mathlib.Algebra.Ring.AddAut",
"Mathlib.Data.Nat.Totient",
"Mathlib.GroupTheory.Divisible",
"Mathlib.Topology.Algebra.IsUniformGroup.Basic",
"Mathlib.Topology.Algebra.Order.Field",
"Mathlib.Topology.IsLocalHomeomorph",
"Mathlib.Topology.Order.T5"
] | Mathlib/Topology/Instances/AddCircle/Defs.lean | continuousAt_toIcoMod | null |
continuousAt_toIocMod (hx : (x : 𝕜 ⧸ zmultiples p) ≠ a) : ContinuousAt (toIocMod hp a) x :=
let h := toIcoMod_eventuallyEq_toIocMod hp a hx
continuousAt_iff_continuous_left_right.2 <|
⟨continuous_left_toIocMod hp a x,
(continuous_right_toIcoMod hp a x).congr_of_eventuallyEq
(h.symm.filter_mono nh... | theorem | Topology | [
"Mathlib.Algebra.Order.ToIntervalMod",
"Mathlib.Algebra.Ring.AddAut",
"Mathlib.Data.Nat.Totient",
"Mathlib.GroupTheory.Divisible",
"Mathlib.Topology.Algebra.IsUniformGroup.Basic",
"Mathlib.Topology.Algebra.Order.Field",
"Mathlib.Topology.IsLocalHomeomorph",
"Mathlib.Topology.Order.T5"
] | Mathlib/Topology/Instances/AddCircle/Defs.lean | continuousAt_toIocMod | null |
AddCircle [AddCommGroup 𝕜] (p : 𝕜) :=
𝕜 ⧸ zmultiples p | abbrev | Topology | [
"Mathlib.Algebra.Order.ToIntervalMod",
"Mathlib.Algebra.Ring.AddAut",
"Mathlib.Data.Nat.Totient",
"Mathlib.GroupTheory.Divisible",
"Mathlib.Topology.Algebra.IsUniformGroup.Basic",
"Mathlib.Topology.Algebra.Order.Field",
"Mathlib.Topology.IsLocalHomeomorph",
"Mathlib.Topology.Order.T5"
] | Mathlib/Topology/Instances/AddCircle/Defs.lean | AddCircle | The "additive circle": `𝕜 ⧸ (ℤ ∙ p)`. See also `Circle` and `Real.angle`. |
coe_nsmul {n : ℕ} {x : 𝕜} : (↑(n • x) : AddCircle p) = n • (x : AddCircle p) :=
rfl | theorem | Topology | [
"Mathlib.Algebra.Order.ToIntervalMod",
"Mathlib.Algebra.Ring.AddAut",
"Mathlib.Data.Nat.Totient",
"Mathlib.GroupTheory.Divisible",
"Mathlib.Topology.Algebra.IsUniformGroup.Basic",
"Mathlib.Topology.Algebra.Order.Field",
"Mathlib.Topology.IsLocalHomeomorph",
"Mathlib.Topology.Order.T5"
] | Mathlib/Topology/Instances/AddCircle/Defs.lean | coe_nsmul | null |
coe_zsmul {n : ℤ} {x : 𝕜} : (↑(n • x) : AddCircle p) = n • (x : AddCircle p) :=
rfl | theorem | Topology | [
"Mathlib.Algebra.Order.ToIntervalMod",
"Mathlib.Algebra.Ring.AddAut",
"Mathlib.Data.Nat.Totient",
"Mathlib.GroupTheory.Divisible",
"Mathlib.Topology.Algebra.IsUniformGroup.Basic",
"Mathlib.Topology.Algebra.Order.Field",
"Mathlib.Topology.IsLocalHomeomorph",
"Mathlib.Topology.Order.T5"
] | Mathlib/Topology/Instances/AddCircle/Defs.lean | coe_zsmul | null |
coe_add (x y : 𝕜) : (↑(x + y) : AddCircle p) = (x : AddCircle p) + (y : AddCircle p) :=
rfl | theorem | Topology | [
"Mathlib.Algebra.Order.ToIntervalMod",
"Mathlib.Algebra.Ring.AddAut",
"Mathlib.Data.Nat.Totient",
"Mathlib.GroupTheory.Divisible",
"Mathlib.Topology.Algebra.IsUniformGroup.Basic",
"Mathlib.Topology.Algebra.Order.Field",
"Mathlib.Topology.IsLocalHomeomorph",
"Mathlib.Topology.Order.T5"
] | Mathlib/Topology/Instances/AddCircle/Defs.lean | coe_add | null |
coe_sub (x y : 𝕜) : (↑(x - y) : AddCircle p) = (x : AddCircle p) - (y : AddCircle p) :=
rfl | theorem | Topology | [
"Mathlib.Algebra.Order.ToIntervalMod",
"Mathlib.Algebra.Ring.AddAut",
"Mathlib.Data.Nat.Totient",
"Mathlib.GroupTheory.Divisible",
"Mathlib.Topology.Algebra.IsUniformGroup.Basic",
"Mathlib.Topology.Algebra.Order.Field",
"Mathlib.Topology.IsLocalHomeomorph",
"Mathlib.Topology.Order.T5"
] | Mathlib/Topology/Instances/AddCircle/Defs.lean | coe_sub | null |
coe_neg {x : 𝕜} : (↑(-x) : AddCircle p) = -(x : AddCircle p) :=
rfl
@[norm_cast] | theorem | Topology | [
"Mathlib.Algebra.Order.ToIntervalMod",
"Mathlib.Algebra.Ring.AddAut",
"Mathlib.Data.Nat.Totient",
"Mathlib.GroupTheory.Divisible",
"Mathlib.Topology.Algebra.IsUniformGroup.Basic",
"Mathlib.Topology.Algebra.Order.Field",
"Mathlib.Topology.IsLocalHomeomorph",
"Mathlib.Topology.Order.T5"
] | Mathlib/Topology/Instances/AddCircle/Defs.lean | coe_neg | null |
coe_zero : ↑(0 : 𝕜) = (0 : AddCircle p) :=
rfl | theorem | Topology | [
"Mathlib.Algebra.Order.ToIntervalMod",
"Mathlib.Algebra.Ring.AddAut",
"Mathlib.Data.Nat.Totient",
"Mathlib.GroupTheory.Divisible",
"Mathlib.Topology.Algebra.IsUniformGroup.Basic",
"Mathlib.Topology.Algebra.Order.Field",
"Mathlib.Topology.IsLocalHomeomorph",
"Mathlib.Topology.Order.T5"
] | Mathlib/Topology/Instances/AddCircle/Defs.lean | coe_zero | null |
coe_eq_zero_iff {x : 𝕜} : (x : AddCircle p) = 0 ↔ ∃ n : ℤ, n • p = x := by
simp [AddSubgroup.mem_zmultiples_iff] | theorem | Topology | [
"Mathlib.Algebra.Order.ToIntervalMod",
"Mathlib.Algebra.Ring.AddAut",
"Mathlib.Data.Nat.Totient",
"Mathlib.GroupTheory.Divisible",
"Mathlib.Topology.Algebra.IsUniformGroup.Basic",
"Mathlib.Topology.Algebra.Order.Field",
"Mathlib.Topology.IsLocalHomeomorph",
"Mathlib.Topology.Order.T5"
] | Mathlib/Topology/Instances/AddCircle/Defs.lean | coe_eq_zero_iff | null |
coe_period : (p : AddCircle p) = 0 :=
(QuotientAddGroup.eq_zero_iff p).2 <| mem_zmultiples p | theorem | Topology | [
"Mathlib.Algebra.Order.ToIntervalMod",
"Mathlib.Algebra.Ring.AddAut",
"Mathlib.Data.Nat.Totient",
"Mathlib.GroupTheory.Divisible",
"Mathlib.Topology.Algebra.IsUniformGroup.Basic",
"Mathlib.Topology.Algebra.Order.Field",
"Mathlib.Topology.IsLocalHomeomorph",
"Mathlib.Topology.Order.T5"
] | Mathlib/Topology/Instances/AddCircle/Defs.lean | coe_period | null |
coe_add_period (x : 𝕜) : ((x + p : 𝕜) : AddCircle p) = x := by
rw [coe_add, ← eq_sub_iff_add_eq', sub_self, coe_period]
@[continuity, nolint unusedArguments] | theorem | Topology | [
"Mathlib.Algebra.Order.ToIntervalMod",
"Mathlib.Algebra.Ring.AddAut",
"Mathlib.Data.Nat.Totient",
"Mathlib.GroupTheory.Divisible",
"Mathlib.Topology.Algebra.IsUniformGroup.Basic",
"Mathlib.Topology.Algebra.Order.Field",
"Mathlib.Topology.IsLocalHomeomorph",
"Mathlib.Topology.Order.T5"
] | Mathlib/Topology/Instances/AddCircle/Defs.lean | coe_add_period | null |
protected continuous_mk' [TopologicalSpace 𝕜] :
Continuous (QuotientAddGroup.mk' (zmultiples p) : 𝕜 → AddCircle p) :=
continuous_coinduced_rng
variable [LinearOrder 𝕜] [IsOrderedAddMonoid 𝕜] | theorem | Topology | [
"Mathlib.Algebra.Order.ToIntervalMod",
"Mathlib.Algebra.Ring.AddAut",
"Mathlib.Data.Nat.Totient",
"Mathlib.GroupTheory.Divisible",
"Mathlib.Topology.Algebra.IsUniformGroup.Basic",
"Mathlib.Topology.Algebra.Order.Field",
"Mathlib.Topology.IsLocalHomeomorph",
"Mathlib.Topology.Order.T5"
] | Mathlib/Topology/Instances/AddCircle/Defs.lean | continuous_mk' | null |
coe_eq_zero_of_pos_iff (hp : 0 < p) {x : 𝕜} (hx : 0 < x) :
(x : AddCircle p) = 0 ↔ ∃ n : ℕ, n • p = x := by
rw [coe_eq_zero_iff]
constructor <;> rintro ⟨n, rfl⟩
· replace hx : 0 < n := by
contrapose! hx
simpa only [← neg_nonneg, ← zsmul_neg, zsmul_neg'] using zsmul_nonneg hp.le (neg_nonneg.2 hx)
... | theorem | Topology | [
"Mathlib.Algebra.Order.ToIntervalMod",
"Mathlib.Algebra.Ring.AddAut",
"Mathlib.Data.Nat.Totient",
"Mathlib.GroupTheory.Divisible",
"Mathlib.Topology.Algebra.IsUniformGroup.Basic",
"Mathlib.Topology.Algebra.Order.Field",
"Mathlib.Topology.IsLocalHomeomorph",
"Mathlib.Topology.Order.T5"
] | Mathlib/Topology/Instances/AddCircle/Defs.lean | coe_eq_zero_of_pos_iff | null |
equivIco : AddCircle p ≃ Ico a (a + p) :=
QuotientAddGroup.equivIcoMod hp.out a | def | Topology | [
"Mathlib.Algebra.Order.ToIntervalMod",
"Mathlib.Algebra.Ring.AddAut",
"Mathlib.Data.Nat.Totient",
"Mathlib.GroupTheory.Divisible",
"Mathlib.Topology.Algebra.IsUniformGroup.Basic",
"Mathlib.Topology.Algebra.Order.Field",
"Mathlib.Topology.IsLocalHomeomorph",
"Mathlib.Topology.Order.T5"
] | Mathlib/Topology/Instances/AddCircle/Defs.lean | equivIco | The equivalence between `AddCircle p` and the half-open interval `[a, a + p)`, whose inverse
is the natural quotient map. |
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