Zeroth homology group for an arbitrary topological space

Question

I´m thinking on a topological space $X$ with pathwise components $C_i$. Since $H_0(C_i) = \mathbb{Z}$ for any $i$. Do you can conclude that the zeroth homology group for any $X$ is ever with the expression $\mathbb{Z} \oplus \overset{i)}{\dots} \oplus \mathbb{Z}$?

Answer 1

Yes! This is exactly right, of course our space might not have finitely many path connected components; however we have the following:

Corresponding to the decomposition of a space $X$ into its path connected components $X_\alpha$ there is an isomorphism of $H_n(X) \simeq \bigoplus_\alpha H_n(X_\alpha)$.

This is true for all $n$. Can you prove it? After we have this, applying to $n=0$ gives your result not only for finite direct sums, but arbitrary sums too!

I would recommend Hatcher's Algebraic topology for much more like this. https://www.math.cornell.edu/~hatcher/AT/AT.pdf
Your question and the proof of this statement are both on page 109.