The Boolean space of higher level orderings

Katarzyna Osiak¹

¹ Institute of Mathematics Silesian University Bankowa 14 40-007 Katowice, Poland

Fundamenta Mathematicae, Tome 196 (2007) no. 2, pp. 101-117.

Résumé

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 $Y$.

Détail

DOI : 10.4064/fm196-2-1

Keywords: ordered field set its orderings topologized make boolean space moreover has shown craven boolean space there exists field homeomorphic beckers higher level ordering generalization usual concept ordering similar ordinary orderings define topology space orderings fixed exact level boolean however main theorem says any boolean space there exists field space orderings fixed exact level which homeomorphic