English
Definition of a finite relative CW construction: given C, D and data describing cells and maps with disjointness and regularity properties, one can form a RelCWComplex on C with D as the relative part.
Русский
Определение конечного относительного CW-конструктора: даны C, D и данные, описывающие клетки и отображения с попарной непересекаемостью и регулярностью; из этого образуется RelCWComplex на C с D как относительная часть.
LaTeX
$$$$ \\text{mkFiniteType}(C,D,\\dots) \\;:\\; \\text{RelCWComplex}(C,D). $$$$
Lean4
/-- If we want to construct a relative CW complex of finite type, we can add the condition
`finite_cell` and relax the condition `mapsTo`. -/
@[simps -isSimp]
def 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 = ball 0 1)
(continuousOn : ∀ (n : ℕ) (i : cell n), ContinuousOn (map n i) (closedBall 0 1))
(continuousOn_symm : ∀ (n : ℕ) (i : cell n), ContinuousOn (map n i).symm (map n i).target)
(pairwiseDisjoint' : (univ : Set (Σ n, cell n)).PairwiseDisjoint (fun ni ↦ map ni.1 ni.2 '' ball 0 1))
(disjointBase' : ∀ (n : ℕ) (i : cell n), Disjoint (map n i '' ball 0 1) D)
(mapsTo :
∀ (n : ℕ) (i : cell n), MapsTo (map n i) (sphere 0 1) (D ∪ ⋃ (m < n) (j : cell m), map m j '' closedBall 0 1))
(closed' :
∀ (A : Set X) (_ : A ⊆ C), ((∀ n j, IsClosed (A ∩ map n j '' closedBall 0 1)) ∧ IsClosed (A ∩ D)) → IsClosed A)
(isClosedBase : IsClosed D) (union' : D ∪ ⋃ (n : ℕ) (j : cell n), map n j '' closedBall 0 1 = C) : RelCWComplex C D
where
cell := cell
map := map
source_eq := source_eq
continuousOn := continuousOn
continuousOn_symm := continuousOn_symm
pairwiseDisjoint' := pairwiseDisjoint'
disjointBase' := disjointBase'
mapsTo n
i := by
use fun m ↦ finite_univ.toFinset (s := (univ : Set (cell m)))
simp only [Finite.mem_toFinset, mem_univ, iUnion_true]
exact mapsTo n i
closed' := closed'
isClosedBase := isClosedBase
union' := union'
