Betti number

Summary

The $k$-th Betti number $b_k(X)$ of a topological space $X$ is the rank of its $k$-th integral singular homology group. Informally, $b_0$ counts the path components, $b_1$ counts independent loops that do not bound, and $b_2$ counts enclosed cavities. Betti numbers are homotopy invariants, and their alternating sum is the Euler characteristic.