## The Boolean space of higher level orderings

### Volume 196 / 2007

Fundamenta Mathematicae 196 (2007), 101-117
MSC: Primary 12D15; Secondary 13J30.
DOI: 10.4064/fm196-2-1

#### Abstract

Let $K$ be an ordered field. The set $X(K)$ of its orderings can be topologized to make it a Boolean space. Moreover, it has been shown by Craven that for any Boolean space $Y$ there exists a field $K$ such that $X(K)$ is homeomorphic to $Y$. Becker's higher level ordering is a generalization of the usual concept of ordering. In a similar way to the case of ordinary orderings one can define a topology on the space of orderings of fixed exact level. We show that it need not be Boolean. However, our main theorem says that for any $n$ and any Boolean space $Y$ there exists a field, the space of orderings of fixed exact level $n$ of which is homeomorphic to $Y$.