Let $A,B$ be two open set of a topological space, $H_1(A\cap B)\to H_1B$ is 1-1 implies $H_1A\to H_1(A\cup B)$ is 1-1, where the homomorphisms is induced by inclusion.
I feel that using the Mayer-Vietoris exact sequence is the key to this question since
$$H_1(A\cap B)\to H_1A\times H_1B\to H_1(A\cup B)$$
is an exact sequence, where the first homomorphism takes $[\gamma]_{A\cap B}\mapsto([\gamma]_A,-[\gamma]_B)$ and the second homomorphism takes $([\gamma]_A,[\delta]_B)\mapsto[\gamma+\delta]_{A\cup B}$. But how should I use these facts to prove the claim?
