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Given $H$ a homology theory, let $f: X \to P$ the unique map from $X$ to a one point space $P$. This induces a map $f_{*}: H_i(X) \to H_{i}(P)$. We define the reduced homology to be $\tilde{H}_i(X, A) = \ker f_{*}$. I would like to show that

Assume $A \ne \emptyset$. Given the inclusion $g : X \to (X, A)$, I would like to prove that $g_{*}H_0(X) = g_*\tilde{H}_0(X)$. One inclusion is clear, but the other is not so clear to me. I've been trying to derive it by examining the induced exact sequences, but I am getting nowhere. Any help would be appreciated.

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This is only true if $A \ne \varnothing$. – Sammy Black Apr 14 '13 at 0:35
I've edited the question. – user72463 Apr 14 '13 at 0:36

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