Finite simplicial complex

A finite simplicial complex is a simplicial complex containing only finitely many simplexes. With the convention that every vertex belongs to a simplex, this is equivalent to having finitely many vertices of a simplicial complex: a finite vertex set has only finitely many subsets.

Its geometric realization is a finite union of compact geometric simplexes, hence compact. The simplicial chain groups have finite bases, so their boundary maps can be represented by finite matrices. This makes simplicial homology a finite linear-algebra calculation after coefficients are chosen.

Three vertices and their three joining edges form a finite complex whose realization is a circle. Adding the filled triangular face produces a finite complex realizing a disk, which is contractible. Thus finiteness makes the data manageable but does not determine the homotopy type.