2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/156240In 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.16 pagesCommutative AlgebraGeneral Topology12D15; 14P05text