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By the excision property of Homology Theory, I know that

$h_{n}(X,X) \cong h_{n}(X-X, X-X) = h_{n}(\phi,\phi)$, since the closure of $X$ in $X$ is equal to the interior of $X$ in $X$ ($X$ is both open and closed in itself).

Based on the axioms, it seems that I cannot conclude anything else about $h_{n}(\phi,\phi)$, which I feel should be $0$. Am I missing something?

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up vote 2 down vote accepted

I think you can easily use the long exact sequence and additivity to show $h_n(\emptyset, \emptyset) \cong 0$

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ok I'll start there! thanks! – roo Apr 11 '12 at 0:01
it worked, thanks a bunch! – roo Apr 11 '12 at 0:07
No problem! Have fun! – John Stalfos Apr 11 '12 at 0:07

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