Betti number

Remarks

When $H_k(X)$ is finitely generated, it is isomorphic to $\Z^{b_k} \oplus T$ with $T$ finite, so $b_k$ is the number of copies of $\Z$. By the universal coefficient theorem, $b_k(X) = \dim_{\Q} H_k(X; \Q)$. Torsion is invisible to Betti numbers.

Example: for the torus $T^2 = S^1 \times S^1$, $H_0 = \Z$, $H_1 = \Z^2$ and $H_2 = \Z$, so $b_0 = 1$, $b_1 = 2$, $b_2 = 1$.

Non-example of a finite Betti number: an infinite discrete space $X$ has $H_0(X)$ free abelian of infinite rank, so $b_0(X) = \infty$. For the real projective plane, $H_1 = \Z/2$ is nonzero but has rank $0$, so $b_1 = 0$.