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Suppose $I$ is some index set and let $\{X_i \}_{i\in I}$ be a collection of topological spaces (as nice as you like them to be). What is known about the (say) singular homology of $X := \prod_{i\in I} X_i$?

If $I$ is finite, there are of course Künneth formulas that relate $H_*(X_i)$ to $H_*(X)$. I am looking for similar tools to compute $H_*(X)$ in the case that the index set is infinite. I am not looking for the most general statement one might make, so please assume additional hypotheses if they are needed.

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