For any left orderable group G, we recall from work of McCleary that isolated
points in the space of left orderings correspond to basic elements in the free
lattice ordered group over G. We then establish a new connection between the
kernels of certain maps in the free lattice ordered group over G, and the
topology on the space of left orderings of G. This connection yields a simple
proof that no left orderable group has countably infinitely many left
orderings. When we take G to be the free group of rank n, this connection sheds
new light on the space of left orderings of the free group: by applying a
result of Kopytov, we show that there exists a left ordering of the free group
whose orbit is dense in the space of left orderings. From this, we obtain a new
proof that the space of left orderings of a free group contains no isolated
points.