# Is there a Mayer-Vietoris sequence for an (uncountably) infinite collection of sets?

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 $$I$$.