  We study the topological space of left-orderings of the braid group, and its
subspace of Nielsen-Thurston orderings. Our main result is that no
Nielsen-Thurston ordering is isolated in the space of braid orderings. In the
course of the proof, we classify the convex subgroups and calculate the
Conradian soul for any Nielsen-Thurston ordering of B_n. We also prove that for
a large class of Nielsen-Thurston orderings, including all those of infinite
type, a stronger result holds: they are approximated by their own conjugates.
On the other hand, we suggest an example of a Nielsen-Thurston ordering which
may not be approximated by its conjugates.
