The space of left orders of a group is either finite or uncountable

TL;DR

This paper proves that the set of all left orderings of a group is either finite or uncountably infinite, using topological methods, correcting previous inaccuracies in earlier work.

Contribution

It establishes a definitive dichotomy for the size of the space of left orders on a group, clarifying the structure of O_G.

Findings

O_G is either finite or uncountably infinite

The topology on O_G is crucial for the proof

Corrects previous inaccuracies in the literature

Abstract

Let G be a group and let O_G denote the set of left orderings on G. Then O_G can be topologized in a natural way, and we shall study this topology to show that O_G can never be countably infinite. This paper retrieves correct parts of the withdrawn paper arXiv:math/0607470.