Theorems · Theorem · algebraic topology

Topology.RelCWComplex.isClosed_of_disjoint_openCell_or_isClosed_inter_closedCell

∀ {X : Type u_1} [t : TopologicalSpace X] {C D : Set X} [inst : Topology.RelCWComplex C D] [T2Space X] {A : Set X},
  A ⊆ C →
    IsClosed (A ∩ D) →
      (∀ (n : ℕ),
          0 < n →
            ∀ (j : Topology.RelCWComplex.cell C n),
              Disjoint A (Topology.RelCWComplex.openCell n j) ∨ IsClosed (A ∩ Topology.RelCWComplex.closedCell n j)) →
        IsClosed A

If for every cell either A ∩ openCell n j is empty or A ∩ closedCell n j is closed then A is closed.

Defined in

Mathlib.Topology.CWComplex.Classical.Basic

Cited by

1 results in Mathlib

Foundations

Depth 162 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

TopologicalSpaceTopology.RelCWComplexT2Space