  Given $s \in (0,1)$, we discuss the embedding of $\mathcal D^{s,p}_0(\Omega)$
in $L^q(\Omega)$. In particular, for $1\le q < p$ we deduce its compactness on
all open sets $\Omega\subset \mathbb R^N$ on which it is continuous. We then
relate, for all q up the fractional Sobolev conjugate exponent, the continuity
of the embedding to the summability of the function solving the fractional
torsion problem in $\Omega$ in a suitable weak sense, for every open set
$\Omega$. The proofs make use of a non-local Hardy-type inequality in $\mathcal
D^{s,p}_0(\Omega)$, involving the fractional torsion function as a weight.
