  A residually finite group $G$ has the Wilson-Zalesskii property if for all
finitely generated subgroups $H,K \leqslant G$, one has $\bar{H} \cap
\bar{K}=\overline{H \cap K}$, where the closures are taken in the profinite
completion $\widehat G$ of $G$. This property played an important role in
several papers, and is usually combined with separability of double cosets. In
the present note we show that the Wilson-Zalesskii property is actually enjoyed
by every double coset separable group. We also construct an example of a LERF
group that is not double coset separable and does not have the Wilson-Zalesskii
property.
