TITLE: Counterexample to $H_n(X)\oplus H_n(X)\cong H_n(X\vee X)$
QUESTION [7 upvotes]: By homology I mean singular homology with coefficients in $\mathbb Z$.
Let $(X,x_0)$ be a based space. It is well known that $H_n(X)\oplus H_n(X)\cong H_n(X\vee X)$ if $X$ is nondegenerately based. Is there a counterexample when this condition doesn't hold?

REPLY [2 votes]: Let $X= C\mathcal{H}$ be the cone on the Hawaiian earring $\mathcal{H}$. Then $H_1(X\vee X)$ is uncountable according to Theorem 2.6 in this paper. Here the wedge sum is formed at the interesting points.
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