  Every left-invariant ordering of a group is either discrete, meaning there is
a least element greater than the identity, or dense. Corresponding to this
dichotomy, the spaces of left, Conradian, and bi-orderings of a group are
naturally partitioned into two subsets. This note investigates the structure of
this partition, specifically the set of dense orderings of a group and its
closure within the space of orderings. We show that for bi-orderable groups
this closure will always contain the space of Conradian orderings---and often
much more. In particular, the closure of the set of dense orderings of the free
group is the entire space of left-orderings.
