English
A RelCWComplex is finite iff the collection of all its cells is finite.
Русский
RelCWComplex конечен тогда и только тогда, когда множество всех клеток конечное.
LaTeX
$$$$ \\text{Finite}(C) \\;\\Longleftrightarrow\\; \\operatorname{Finite}\\left(\\sum_{n} \\text{cell}(C,n)\\right). $$$$
Lean4
/-- A CW complex that was constructed using `RelCWComplex.mkFinite` is finite. -/
theorem 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))
(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))
(isClosedBase : IsClosed D) (union' : D ∪ ⋃ (n : ℕ) (j : cell n), map n j '' closedBall 0 1 = C) :
letI :=
mkFinite C D cell map eventually_isEmpty_cell finite_cell source_eq continuousOn continuousOn_symm
pairwiseDisjoint' disjointBase' mapsTo isClosedBase union'
Finite C :=
letI :=
mkFinite C D cell map eventually_isEmpty_cell finite_cell source_eq continuousOn continuousOn_symm pairwiseDisjoint'
disjointBase' mapsTo isClosedBase union'
{ eventually_isEmpty_cell := eventually_isEmpty_cell
finite_cell := finite_cell }
