  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 answer
three conjectures. In particular we shall show that O_G can never be countably
infinite. Furthermore in the case G is a countable nonabelian free group, we
shall show that O_G is homeomorphic to the Cantor set and that the positive
cone of a left order on G is not finitely generated. Generalizations to locally
indicable groups will also be considered.
