# Intersection of two open connected sets in the plane

Assume $$O_1$$ and $$O_2$$ are two open connected sets of $$\mathbb{C}$$ such that $$O_1\cup O_2=\mathbb{C}$$.

Is it true that $$O_1\cap O_2$$ is connected ?

Yes. For open subsets of $$\mathbb{C}$$ connectedness agrees with pathwise connectedness (this is true for any locally pathwise connected space).

The Mayer-Vietoris-sequence for reduced singular homology gives us an exact sequence

$$H_1(\mathbb{C}) \to \tilde{H}_0(O_1 \cap O_2) \to \tilde{H}_0(O_1) \oplus \tilde{H}_0(O_2) .$$

But $$H_1(\mathbb{C}) = 0$$ and $$\tilde{H}_0(O_k) = 0$$, hence $$\tilde{H}_0(O_1 \cap O_2) = 0$$. This implies that $$O_1 \cap O_2$$ has only one path component.

Note the $$H_0(X)$$ is always a free abelian group whose basis can be identified with the set of path components of $$X$$.

This generalizes as follows: Let $$X$$ be connected with $$H_1(X) = 0$$ and $$O_1,O_2$$ two open path connected subsets such that $$O_1 \cup O_2 = X$$. Then $$O_1 \cap O_2$$ is path connected.

• Very nice, Thanks ! – user111 Sep 29 '18 at 14:19