 Simplicial complex

An abstract simplicial complex is a pair $K=(V,\Sigma )$, where $V$ is a set of vertices of a simplicial complex and $\Sigma$ is a collection of finite nonempty subsets, called simplexes. Every singleton $\{v\}$ belongs to $\Sigma$, and every nonempty subset of a simplex is a simplex. Its dimension is the supremum of $|\sigma |-1$ over $\sigma \in \Sigma$. Some conventions also include an empty simplex; it contributes no points to the geometric realization.

A simplicial complex is a mathematical structure used in algebraic topology and combinatorial mathematics to study spaces and their properties. It is a way of building up a geometric object from simpler building blocks called simplices. ### Definition of a Simplicial Complex A simplicial complex \( K \) is a set of simplices that satisfies two conditions: 1. **Non-emptiness**: The empty set is in \( K \).