Spaces of $\mathbb{R}$ - places of rational function fields

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 $\mathbf{\text{R}}(Y)$ (where $\mathbf{\text{R}}$ is any real closure of $\mathbb{R}(X)$) is not metrizable space. Thus the space $M(\mathbb{R}(X,Y))$ is not metrizable, too.