  For each $n\geq2$ we construct an unbounded closed pseudoconcave complete
pluripolar set $\mathcal E$ in $\mathbb C^n$ which contains no analytic variety
of positive dimension (we call it a \textit{Wermer type set}). We also
construct an unbounded strictly pseudoconvex domain $\Omega$ in $\mathbb C^n$
and a smooth $CR$ function $f$ on $\partial\Omega$ which has a single-valued
holomorphic extension exactly to the set $\bar\Omega\setminus\mathcal E$.}
