The Boolean space of orderings of a field

Craven, Thomas C.

doi:10.1090/S0002-9947-1975-0379448-X

The Boolean space of orderings of a field
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by Thomas C. Craven PDF

Trans. Amer. Math. Soc. 209 (1975), 225-235 Request permission

Abstract:

It has been pointed out by Knebusch, Rosenberg and Ware that the set $X$ 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 $W(a) = \{ < {\text { in }}X|a < 0\}$ the Harrison subbasis of $X$. This subbasis is closed under symmetric difference and complementation. In this paper it is proved that, given any Boolean space $X$, there exists a formally real field $F$ such that $X$ is homeomorphic to the space of orderings on $F$. 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.

References

Éléments de mathématique

Les structures fondamentales de l’analyse

Algèbre

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