Simplicial Complex

A combinatorial structure given by a collection K of finite sets (called simplices) on a vertex set V such that every singleton vertex is in K and K is closed under taking nonempty subsets: if σ∈K and τ⊆σ then τ∈K. Simplices are organized by dimension (|σ|−1).

A combinatorial object consisting of a set of vertices and a collection of finite subsets (simplices) closed under taking nonempty faces: if a simplex is included then every nonempty subset of its vertices is also included.

An (abstract) simplicial complex K on a vertex set V is a family of finite subsets (called simplices) of V closed under taking nonempty subsets: if σ ∈ K and τ ⊆ σ then τ ∈ K. Geometric realizations glue simplices along faces to model piecewise-linear topology.