For two connected open sets $U$ and $V$ of $S^n(n\geq 2)$,if $U \cup V=S^n$,then $U \cap V$ is also connected.
1.I have no idea how to use (reduced) singular homology groups to describe connectedness of topological spaces.I just know $ {H_0 ^\sim}(X)=0$ iff X is path-connected.
2.As for the method of using degree map of $S^n$, I have trouble in constructing a good map.To be honest, again, I don't know how to describe connectedness.
Any hint will be greatly appreciated, thanks!
First, by @Nicolas' comment connected and path-connected are equivalent in this context. As in @Lord's comment (and hence marking this answer cw), the Mayer-Vietoris sequence (in reduced homology) ends with $$H_1(S^n) \to \tilde H_0(A \cap B) \to \tilde H_0(A) \oplus \tilde H_0(B) \to \tilde H_0(S^n)$$ Since $S^n$ is simply connected and assuming $A$ and $B$ are connected, this is an exact sequence $$0 \to \tilde H_0(A \cap B) \to 0 \to 0$$ and hence $\tilde H_0(A \cap B) = 0$, which means $A \cap B$ is connected.