Alphabeta Math
DefinitionDefinition: Literature-sourcedProof: Not applicablejudge pass (gpt-5.6-terra)audited 2026-09-04
How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

  • Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
  • AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
  • AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

Subcomplexes, closures, stars, and links in a simplicial complex

Definition

Let (V,K) be an abstract simplicial complex.

A subcomplex of K is a subfamily L⊆K that is itself closed under taking subsets.

If σ∈K, its closure is the subcomplex σ‾:={τ∈K:τ⊆σ}. More generally, if A⊆K, its closure is A‾:={τ∈K:τ⊆σ for some σ∈A}.

For a simplex σ∈K, the closed star of σ in K is St⁡K(σ):={τ∈K:σ∪τ∈K}, and the open star is the union, after realization, of the relative interiors of simplices ρ∈K with σ⊆ρ.

The link of σ in K is Lk⁡K(σ):={τ∈K:τ∩σ=∅ and σ∪τ∈K}.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · one level

1 result within one dependency step of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources