= Solution
Let $C$ be a component of the <open set> $X\subseteq\mathbb R$. Being a connected subset of the real line, $C$ is an <interval>. For any $x\in C$, openness of $X$ gives $\epsilon>0$ with $(x-\epsilon,x+\epsilon)\subseteq X$. This <interval> is connected and meets $C$, so their union is connected. Maximality of $C$ makes the entire <interval> lie in $C$. Thus every point of $C$ is interior, so \b[each component is an open <interval>], allowing unbounded endpoints.
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