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 $L \subseteq K$ that is itself closed under taking subsets.

If $σ \in K$ , its closure is the subcomplex $\overline{σ} : = {τ \in K : τ \subseteq σ} .$ More generally, if $A \subseteq K$ , its closure is $\overline{A} : = {τ \in K : τ \subseteq σ for some σ \in A} .$

For a simplex $σ \in K$ , the closed star of $σ$ in $K$ is ${St}_{K} (σ) : = {τ \in K : σ \cup τ \in K},$ and the open star is the union, after realization, of the relative interiors of simplices $ρ \in K$ with $σ \subseteq ρ$ .

The link of $σ$ in $K$ is ${Lk}_{K} (σ) : = {τ \in K : τ \cap σ = \emptyset and σ \cup τ \in K} .$

Depends on

An abstract simplicial complex

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

Definition

Depends on

Dependency tree · 1 level

Sources