  We classify $C$-orderable groups admitting only finitely many $C$-orderings.
We show that if a $C$-orderable group has infinitely many $C$-orderings, then
it actually has uncountably many $C$-orderings, and none of these is isolated
in the space of $C$-orderings. As a relevant example, we carefully study the
case of Baumslag-Solitar's group B(1,2). We show that B(1,2) has four
$C$-orderings, each of which is bi-invariant, but its space of left-orderings
is homeomorphic to the Cantor set.
