= Solution
Suppose a <connected> subset $C\subseteq F$ contains distinct points $a,b$. The distance function $d_a(x)=|x-a|$ is a <Lipschitz map> with constant one, by the <reverse triangle inequality>. Its image $d_a(C)$ is <connected> and contains both $0$ and $|b-a|$, so it contains the whole interval $[0,|b-a|]$. Monotonicity and the <Lipschitz> inequality for <Hausdorff dimension> imply
$$
1=\dim_H[0,|b-a|]\leq\dim_H d_a(C)\leq\dim_H C\leq\dim_H F<1,
$$
a contradiction. Thus every nonempty <connected> subset of $F$ is a single point: \b[$F$ is <totally disconnected>]. The argument is the <Hausdorff dimension of a connected set> bound, and does not require $F$ to be <closed>.
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