Betti number

This article describes an invariant of topological spaces that depends only on its homology groups

Definition

Given a topological space $X$, the $n^{th}$ Betti number of $X$, denoted $b_{n}(X)$, is a nonnegative integer defined as follows:

1. It is the free rank of the $n^{th}$ singular homology group $H_{n}(X;\mathbb {Z} )$, where free rank refers to the rank of the torsion-free part. This makes sense if the $n^{th}$ singular homology group is finitely generated, or more generally, if its quotient by its torsion subgroup is finitely generated.
2. It is the dimension of the $n^{th}$ singular homology group $H_{n}(X;\mathbb {Q} )$ as a vector space over $\mathbb {Q}$.