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.

An abstract simplicial complex

Definition

An abstract simplicial complex is a pair $(V, K)$ consisting of a set $V$ of vertices and a family $K$ of finite subsets of $V$ such that:

$\emptyset \in K$ ;
if $σ \in K$ and $τ \subseteq σ$ , then $τ \in K$ ;
every singleton ${v}$ with $v \in V$ lies in $K$ .

The elements of $K$ are the simplices of the complex. If $σ \in K$ is nonempty, its dimension is $\dim σ : = ∣ σ ∣ - 1$ , and the empty simplex has dimension $- 1$ .

Used by

A simplicial map and its geometric realization Definition
An orientation of a simplex Definition
Local finiteness, finiteness, and finite dimensionality of a simplicial complex Definition
Subcomplexes, closures, stars, and links in a simplicial complex Definition
The geometric realization of an abstract simplicial complex Definition

Dependency tree · 0 levels

Nothing. This result depends on no other item in the library.

Sources

Allen Hatcher, Algebraic Topology (standard reference, not scraped)
Vidit Nanda, Computational Algebraic Topology, Lecture 01: Complexes (standard reference, not scraped)