The following comes from Conway's Complex Analysis book. Let $X = \{z : |z| \le 1 \} \cup \{z : |z-2|<1\} \subset \mathbb{C}$. I suspect that is is connected, but I am having difficulty demonstrating this. I could use some hints, tips, etc.
Here is the definition of connectedness that I am working with:
A metric space $(X,d)$ is connected if the only subsets of $X$ that are both open and closed are $\emptyset$ and $X$. If $A \subseteq X$, then $A$ is connected if $(A,d)$ is connected.
Suppose to the contrary that the set is the union of two disjoint open sets. Take the open set (call $A$) that contains $z=1$. Argue that any open set contain this point has to intersect both balls. Now the other open set (call $B$) intersects one of the two balls non-trivally, call this ball $X$. Now $X\cap A$ and $X\cap B$ are (relatively) open disjoint subsets of $X$ that are both non-empty and cover $X$, implying $X$, a ball, is non-connected.