In the paper an answer to a problem "When different orders of R(X) (where R is a real closed field) lead to the same real place ?" is given. We use this result to show that the space of $\mathbb R$-places of the field $\textbf{ R}(Y)$ (where \textbf{ R} is any real closure of $\mathbb R(X)$) is not metrizable space. Thus the space $M(\mathbb R(X,Y))$ is not metrizable, too.
Categories: math.AC math.GN