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 $X$ is defined as the rank of the $n^{th}$ singular homology group (rank here is as in the rank of a free Abelian group). Here, we take the singular homology theory over $\mathbb{Z}$.