= 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>.
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