  We study spaces $M(R(y))$ of $\R$-places of rational function fields $R(y)$
in one variable. For extensions $F|R$ of formally real fields, with $R$ real
closed and satisfying a natural condition, we find embeddings of $M(R(y))$ in
$M(F(y))$ and prove uniqueness results. Further, we study embeddings of
products of spaces of the form $M(F(y))$ in spaces of $\R$-places of rational
function fields in several variables. Our results uncover rather unexpected
obstacles to a positive solution of the open question whether the torus can be
realized as a space of $\R$-places.
