Singular homology groups of a disjoint union of spaces

Question

Is it true, for a disjoint countable union $X$ of topological spaces $X_j$, that $\bigoplus_jH_n(X_j)$ is isomorphic to $H_n(X)$?

Here $\bigoplus_jH_n(X_j)$ denotes the group of elements $(x_1,…,x_j,…)$ under pointwise addition. However for the groups to be isomorphic I would like $\bigoplus_jH_n(X_j)$ to denote the group of elements $(x_1,…,x_j,…)$ such that only finitely many elements are nonzero.

In this case the mapping $\varphi$, given by $\varphi(x_1,…,x_j…)=\sum \iota_j^*(x_j)$ is well defined since it is a finite sum of elements in $H_n(X)$. Here $\iota_j$ is the inclusion from $X_j$ to $X$. My intuition tells me that $\varphi$ is an isomorphism. Is this correct?

For a finite union everything is clear, it is the countable union I’m wondering about. — spinosarus123, Aug 31 '21 at 13:19
But for the ”universal map” to be well defined only finitely many elements can be nonzero right? Because otherwise we end up with a sum of infinitely many elements in $H_p(\bigsqcup_{\alpha \in J} X_\alpha)$. And we are only allowed finite linear combination because it is (quotient of) a free abelian group. — spinosarus123, Aug 31 '21 at 13:35
Yes. But isn't that the definition of direct sum? That only finitely many terms are non-zero? — Arctic Char, Aug 31 '21 at 13:50
Oh, in that case that’s where my confusion was. Thank you! — spinosarus123, Aug 31 '21 at 13:52
You may consider closing this as a duplicate of that question then (click "close" then "duplicate") — Arctic Char, Aug 31 '21 at 13:54

score 1 · Accepted Answer · answered Aug 31 '21 at 13:55

It is irrelevant whether the $X_j$ form a countable family or not. We always have $$H_n(X) \approx \bigoplus_jH_n(X_j) .$$

Your mapping $\varphi$ is in fact an isomorphism. Note that the direct sum $\bigoplus_jH_n(X_j)$ is defined as the group of elements $(x_l)_{j\in J}$ such that only finitely many elements $x_j$ are nonzero. See Direct sum of abelian groups.

Singular homology groups of a disjoint union of spaces

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