The structure of spaces of $\mathbb{R}$-places of rational function fields over real closed fields

Abstract

For arbitrary real closed fields $R$, we study the structure of the space $M(R(y))$ of $\mathbb{R}$-places of the rational function field in one variable over $R$ and determine its dimension to be 1. We determine small subbases for its topology and discuss a suitable metric in the metrizable case. In the case of non-archimedean $R$, we exhibit the rich variety of homeomorphisms of subspaces that can be found in such spaces.