  We investigate the form of the closure of the smooth, compactly supported
functions $C_{c}^{\infty}(\Omega)$ in the weighted fractional Sobolev space
$W^{s,p;\,w,v}(\Omega)$ for bounded $\Omega$. We focus on the weights $w,\,v$
being powers of the distance to the boundary of the domain. Our results depend
on the lower and upper Assouad codimension of the boundary of $\Omega$. For
such weights we also prove the comparability between the full weighted
fractional Gagliardo seminorm and the truncated one.
