In Allen Hatcher's Algebraic Topology, Van Kampen's theorem is stated for a (possibly uncountably infinite) collection of path-connected open sets $A_\alpha$ whose union is some topological space $X$.

Mayer-Vietoris sequences however, which are the analog for homology of Van Kampen's theorem for the fundamental group, are only stated to exist for pairs of subspaces $A, B \subset X$, such that $X$ is the union of the interiors of $A$ and $B$.

To be precise, I was wondering whether for a collection of subspaces $(A_\alpha)_{\alpha\in I}$, such that $\bigcup\limits_{\alpha\in I} A_\alpha = X$, there exists an exact sequence $$\dots\longrightarrow H_n(\cap_{\alpha\in I} A_\alpha)\longrightarrow \oplus_{\alpha\in I} H_n(A_\alpha)\longrightarrow H_n(X)\longrightarrow H_{n-1}(\cap_{\alpha\in I} A_\alpha)\longrightarrow\dots$$

given an uncountable or countably infinite index set $I$.


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