The Boolean space of orderings of a field

Abstract

It has been pointed out by Knebusch, Rosenberg and Ware that the set X X of all orderings on a formally real field can be topologized to make a Boolean space (compact, Hausdorff and totally disconnected). They have called the sets of orderings W ( a ) = { >  in  X | a > 0 } W(a) = \{ > {\text { in }}X|a > 0\} the Harrison subbasis of X X . This subbasis is closed under symmetric difference and complementation. In this paper it is proved that, given any Boolean space X X , there exists a formally real field F F such that X X is homeomorphic to the space of orderings on F F . Also, an example is given of a Boolean space and a basis of clopen sets closed under symmetric difference and complementation which cannot be the Harrison subbasis of any formally real field.