= Solution
The inclusion of $X$ into $\mathbb R$ is <continuous>, so a <connected component> $K$ of $X$ is an <interval> in $\mathbb R$. For $x\in K$, openness of $X$ gives $\varepsilon>0$ with $(x-\varepsilon,x+\varepsilon)\subseteq X$. This <interval> is <connected> and meets $K$, so their union is <connected> and maximality forces the <interval> into $K$. Thus $K$ is open in $\mathbb R$. A nonempty open <connected subset> of the line is an <open interval>, allowing an infinite endpoint or the whole line. Hence \b[every component is an <open interval>].
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