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

TL;DR

This paper investigates the conditions under which different orderings of rational function fields over real closed fields yield the same real place, and demonstrates that the space of real places of certain rational function fields is not metrizable.

Contribution

It provides a characterization of when different orders lead to the same real place and shows the non-metrizability of the space of real places for specific rational function fields.

Findings

Different orders can produce the same real place under certain conditions.

The space of real places of R(Y) is not metrizable.

The space M(R(X,Y)) is also not metrizable.

Abstract

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.