Let $A$ and $B$ be two open sets in $\mathbb{R}^n$ ($n>1$), with $A \cup B=\mathbb{R}^n$ and $A\cap B\not=\emptyset$. There is a version of the Mayer-Vietoris sequence for reduced homology that implies: $$...\widetilde{H}_1(\mathbb{R}^n,\mathbb{Z})\to \widetilde{H}_0(A\cap B,\mathbb{Z})\to \widetilde{H}_0(A,\mathbb{Z})\oplus\widetilde{H}_0(B,\mathbb{Z})\to\widetilde{H}_0(\mathbb{R}^n,\mathbb{Z})... $$is exact. Do you know any reference for this version?
The answer should be in the relative homology groups (Hatcher pg 115) and the relative version of Mayer-Vietoris sequence (Hatcher pg 150). This is suggested by the discussion mentioned by
HallaSurvivor, but I don't see clearly!