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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 $R$ realclosed and satisfying a natural condition, we find embeddings of $M(R(y))$ in$M(F(y))$ and prove uniqueness results. Further, we study embeddings ofproducts of spaces of the form $M(F(y))$ in spaces of $\R$-places of rationalfunction fields in several variables. Our results uncover rather unexpectedobstacles to a positive solution of the open question whether the torus can berealized as a space of $\R$-places.

Embedding theorems for spaces of $\R$-places of rational function fields and their products