I want to
Find a space X which is covered by finitely many open sets $U_i$ such that $H_n(U_i)$ is finitely generated for all n but $H_n(X)$ is not finitely generated.
I know how to produce Moore spaces for finitely generated Abelian groups and the first guess of not fg group that comes to mind is $\mathbb Q$. But I can't really produce spaces with homnology $\mathbb Q$.
Any help?