Given a countable union of spaces $$X=\bigcup_n Y_n$$ such that all intersections $$\bigcap_{i\in I}Y_i, \vert I\vert\ge 2$$ are contractible (or weaker just have trivial homology).
Is it true that in homology (in degrees $*\ge 2$) $$H_*(X)=\bigoplus_n H_*(Y_n)$$ is the direct sum?
The infinite earring space should be a counterexample: with the usual description as a union of a family of circles of decreasing radii, the intersection of any set of those circles is a point and hence contractible, but the homology of the whole space is larger than the direct sum. The Wikipedia page gives a description.
There are higher dimensional analogues which are also counterexamples: see the paper by Barratt and Milnor, https://www.ams.org/journals/proc/1962-013-02/S0002-9939-1962-0137110-9/.
