Jordan curve theorem (bit generalized one)
Let $C_1$ and $C_2$ be closed connected subsets of $S^2$ whose intersection consists of two points. If neither $C_1$ nor $C_2$ separates $S^2$, then $C_1\cup C_2$ separates $S^2$ into precisely two components.
I think the above theorem can be generalized further and below is the statement I formulated:
Let $C_1$ and $C_2$ be closed connected subsets of $S^2$ whose intersection is a disjoint union of two nonempty connected compact subsets. If neither $C_1$ nor $C_2$ separates $S^2$, then $C_1\cup C_2$ separates $S^2$ into precisely two components.
Is this true? If so how do I prove this? Or else if this is false, what would be a counterexample?
You can show that $H_1(S^2\setminus C_1\cap C_2)\ne 0$ using Mayer–Vietoris sequence, and then use Mayer–Vietoris sequence again for the pair $S^2\setminus C_1$, $S^2\setminus C_2$ and show that rank of $H_0(S^2\setminus(C_1\cup C_2))$ is at least two.