Simplicial complexes

The 0,1,2,3-dimensional simplices are named: vertex, edge, triangle, tetrahedron.

Any subset of affinely independent points again defines a simplex.

A face $\tau$ of the simplex $\sigma =\mathrm{conv}\{{𝒖}_0,\dots ,{𝒖}_k\}$ is the convex hull of some subset of the points, $\tau ⩽\sigma$. It is a proper face if the subset is not the entire set of points, $\tau <\sigma$, and $\sigma$ is said to be the coface of $\tau$.

The union of all proper faces is the boundary of $\sigma$, denoted as $\mathrm{bd}\left(\sigma \right)$. The interior is the complement of the boundary, $\mathrm{int}\left(\sigma \right)=\sigma -\mathrm{bd}\left(\sigma \right)$. A point $𝒙={\sum }_{i=0}^k{\lambda }_i{𝒖}_i\in \sigma$ is in the interior, $𝒙\in \mathrm{int}\left(\sigma \right)$ if ${\lambda }_i>0$. The point $𝒙$ belongs to the interior of the face spanned by the points for which ${\lambda }_i>0$.

Definition. A simplicial complex is a finite collection $𝒦$ of simplices such that $\sigma \in 𝒦$ and $\tau ⩽𝒦$ implies $\tau \in 𝒦$, and $\sigma ,{\sigma }_0\in 𝒦$ implies $\sigma \cap {\sigma }_0$ is either empty or a face of both.

Restated, $𝒦$ is closed under taking faces and has no improper intersections.