 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 implya contradiction. Thus every nonempty connected subset of $F$ is a single point: $F$ is totally disconnected. The argument is the Hausdorff dimension of a connected set bound, and does not require $F$ to be closed.