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Abstract simplicial subcomplex

Definitions

A subcomplex $K'=(V',\Sigma')$ of a simplicial complex $K=(V, \Sigma)$ is a simplicial complex where $V' \subset V$ and $\Sigma' \subset \Sigma$

Properties

In the topological realisation of a simplicial complex, the simplicial subcomplexes are closed.

Examples

Given a set of vertices $V' \subset V$ of a s.c. $K$, the subcomplex spanned by $V'$ is the subcomplex with vertex set $V'$, and simplices $\Sigma' = \{\sigma \in \Sigma: \sigma \subseteq V'\}$.

Given a vertex $v$ of a s.c. $K$, the link of $v$ is the subcomplex lk$(v)$ with vertex set $\{w \in V \setminus \{v\}: \{v, w\} \in \Sigma\}$, and simplices $\sigma \setminus \{v\}$ such that $\sigma \in \Sigma$.