Is the union of two tangent open disks a connected set?

Question

I have a problem defining whether the set $A$ that we get as the union of two tangent open disks is an open set. On the one hand we can select two sets: $B$ as the first disk and $C$ as the second one. Then $B\cup C=A$ ,$B\cap C=\emptyset$ ,$B\neq \emptyset$ , $C\neq \emptyset$ and $B$ does not contain any limt points of $C$ neither does $C$ contain any limit points of $B$.Taking this into consideration we can tell that $A$ is not a connected set. However using another definition of connected sets we can find a sequence of n points: $z_k$ such that $|z_{k-1}-z_k|<r , \quad \forall r>0$ through which any points $\alpha,\beta\in A$ are connected even for the case that the two points belong each to a different disk and so $A$ is considered a connected set. What is my mistake that makes this contradiction arise. Any help is appreciated.

Answer 1

As you shown, $A$ is not connected but it is well-chained. There is no contradiction: every connected metric space is well-chained but a well-chained metric space is not necessarily connected (think of $\mathbb Q$, or $\mathbb R^*$). What is true (but out of subject here) is that every compact well-chained metric space is connected.