skeletonLT — Mathlib

English

Let X be a topological space with a relative CW-structure on C ⊆ X relative to D ⊆ X. For each n ∈ ℕ ∪ {∞}, skeletonLT(C, D; n) is the subcomplex of C generated by the base D together with all closed cells with index m < n.

Русский

Пусть X — топологическое пространство, на C ⊆ X задана относительная CW-структура относительно D ⊆ X. Для каждого n ∈ ℕ ∪ {∞} скелетой skeletonLT(C, D; n) является подслоями (подкомплексом) C, порожденный базой D и всеми замкнутыми клетками индекса m < n.

LaTeX

$$$\\mathrm{skeletonLT}(C,D;n)=\\mathrm{Subcomplex}\\bigl(C,\,D\\cup\\bigcup_{m\\in\\mathbb{N},\,m

Lean4

/-- 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 derive the statement about `skeleton`. -/
@[simps! -isSimp, irreducible]
def 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_iUnion₂_of_subset l hi
      exact subset_iUnion _ _)
    (by
      rw [← RelCWComplex.iUnion_openCell_eq_iUnion_closedCell]
      congrm D ∪ ?_
      apply iUnion_congr fun m ↦ ?_
      rw [iUnion_subtype, iUnion_comm]
      rfl)

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