I have the following question
If $ X=A\cup B$ where $A$ and $B$ are open non empty sets and $A\cap B = \emptyset$ then $H_n(X) \cong H_n(A) \oplus H_n(B)$.
My attempt is to prove this using Mayer-Vietoris theorem as the following:
$$\cdots \rightarrow H_n(A \cap B)\rightarrow H_n(A) \oplus H_n(B) \rightarrow H_n(X) \rightarrow H_{n-1}(A \cap B) \rightarrow\cdots$$
But $H_n(A \cap B ) = 0$ for al $n \geq 0$ so we have the following exact sequence
$$\cdots \rightarrow 0 \rightarrow H_n(A) \oplus H_n(B) \rightarrow H_n(X) \rightarrow 0 \rightarrow \cdots$$ so the proof is finished.
Is it right?