Computational example: Using the zeroth homology group to find the number of path components

Question

I'm studying singular homology, and I have a question about $H_0 (X)$, the zeroth homology group of a space $X$.

I understand that the Betti number of $H_0 (X)$ equals the number of the path components of $X$.

But my question is: is this information useful to compute the number of path components of $X$? If so, then can you give me a computational example? It makes me reluctant to use $H_0(X)$ to find just the number of path components of $X$. The answer would be great if using $H_0(X)$ is a must or the easiest way.

Thank you.

Answer 1

In fact $H_0(X)$ is a free abelian group whose rank (= Betti number) equals the number of the path components of $X$. But there is no way to compute $H_0(X)$ without previously determining the number of the path components of $X$.

Thus the number of path components of $X$ is needed to compute $H_0(X)$, but not conversely.