Simplicial complex

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 $k+1$ is termed a $k$-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.