What are all of the connected subsets of $\mathbb{Q}$?
Solution 1:
Let $A\subseteq\mathbb Q$ with at least two elements, thus $x\in A$, $y\in A$, and $x\ne y$. Let $z$ be an irrational such that $x\lt z\lt y$. Then $A=A_+\cup A_-$ with $A_+=A\cap(z,+\infty)$ and $A_-=A\cap(-\infty,z)$. Since $A_+$ and $A_-$ are two disjoint nonempty sets and are open in $A$, $A$ is not connected.