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I have written a proof for the following and would like you to correct me if I made any mistakes, thanks in advance:

claim: $X = \sqcup_i X_i$ then $H_q (X) \cong \oplus_i H_q (X_i)$

proof: Proof of case $X = A \sqcup B$, the general case follows by induction.

By the Mayer-Vietoris theorem the following sequence is exact:

$$ \dots \xrightarrow{k_\ast} H_{n+1}(X) \xrightarrow{\partial_\ast} H_n(A \cap B) \xrightarrow{(i_\ast, j_\ast)} H_n(A) \oplus H_n(B) \xrightarrow{k_\ast} H_n(X)\xrightarrow{\partial_\ast} \dots$$

Then $A \cap B = \emptyset \implies H_n(A \cap B) = 0 \implies \partial_\ast = 0$

Then $k_\ast$ is injective because $ker k_\ast = im \partial_\ast = 0$ and $k_\ast $ is surjective because $im k_\ast = ker \partial_\ast = H_n(X)$

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    Well, yes, *if* you take Mayer-Vietoris for granted. However, you should probably try and prove this directly from the definition of homology... Suggestion: go through Hatcher's proof of Mayer-Vietoris and extract a direct argument. This would probably be more illuminating than what you did.2011-08-08
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    If your collection is not countable, you will need transfinite induction.2011-08-08
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    Maybe you can argue that if a cycle bounds in one of the $A_i$'s, then it will bound in the disjoint union, and, conversely, a trivial cycle will also be trivial in the union.2011-08-08
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    @Theo, ok, I'll do that.2011-08-08
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    @gary, hey thanks for the hint!2011-08-08
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    You could also make an analogous observation about the singular (or simplicial, or $\Delta$) chain complex for the disjoint union and work from there.2011-08-08
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    @wckronholm: thanks for the hint!2011-08-08
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    @MattN. you can tell me how to solve now you have the answer, please, i also have a question, $A$ and $B$ are closed so how you can use the Mayer-Vietoris theorem ? Thank you2014-08-31
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    @Vrouvrou Sorry this is way too long ago and I think I never really understood Mayer-Vietoris, sadly.2014-09-01

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Wouldn't be easier to argue like this?

Since $\Delta^n$ is connected, then the image of every continuous map $\sigma : \Delta^n \longrightarrow \bigsqcup_{\alpha \in J} X_\alpha$ must be contained in some $X_\alpha$: $\sigma (\Delta^n) \subset X_\alpha$.

I didn't check the details, but I think that this would say that you can find an inverse to the universal map

$$ \bigoplus_{\alpha \in J} H_p(X_\alpha) \longrightarrow H_p(\bigsqcup_{\alpha \in J} X_\alpha) $$

induced by the inclusions $X_\alpha \longrightarrow \bigsqcup_{\alpha \in J} X_\alpha$. And this would be true for any set of indices $J$.

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    I am confused; did the OP specify he is working with singular or simplicial homology?2011-08-08
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    The OP didn't specify, but it's really the same argument either way.2011-08-08