# Simplicial complex

From Topospaces

## Definition

A (finite) simplicial complex is a finite collection of simplies in some Euclidean space such that:

- If any simplex belongs to the complex, so do all its faces
- For any two simplices in the complex, either they do not intersect, or their intersection is a common face of both.

Here a **simplex** is the set of all convex combinations of a finite affine independent subset of the Euclidean space, and a **face** of a simplex is simply a subset of it. The elements of the subset are often called the **vertices** of the simplex, and a simplex of size is termed a -simplex.

The **underlying space** of a simplicial complex is defined as the union of all its simplices. A space that can be expressed as the underlying space of a simplicial complex is termed a polyhedron, and such an expression i s termed a triangulation.