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.

Local finiteness, finiteness, and finite dimensionality of a simplicial complex

Definition

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

The complex is finite if $K$ has only finitely many simplices.

It is locally finite if each vertex $v \in V$ lies in only finitely many simplices of $K$ .

It is finite dimensional if there is an integer $N \geq 0$ such that every simplex of $K$ has dimension at most $N$ .

These are distinct conditions: finite implies locally finite and finite dimensional, but local finiteness and finite dimensionality do not imply finiteness, and finite dimensionality does not imply local finiteness.

Depends on

An abstract simplicial complex

Used by

Finite dimensional does not imply finite or locally finite Counterexample
The simplicial Euler characteristic Definition
A finite simplicial complex has a compact Hausdorff realization Proposition

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

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