  Several results concerning pairs of polynomially convex sets whose union is
not even rationally convex are given. It is shown that there is no restriction
on how two spaces can be embedded in some $\C^N$ so as to be polynomially
convex but have nonrationally convex union. It is shown that there exist two
disjoint polynomially convex Cantor sets in $\C^3$ whose union is not
rationally convex. The analogous assertion for arcs is also established. As an
application it is shown that every simple closed curve in $\C^N$, $N\geq 3$,
can be approximated uniformly by locally polynomially convex simple closed
curves that are not rationally convex.
