  In this paper, we continue the investigation of topological properties of
unbounded norm (un-)topology in normed lattices. We characterize separability
and second countability of un-topology in terms of properties of the underlying
normed lattice. We apply our results to prove that an order continuous Banach
function space $X$ over a semi-finite measure space is separable if and only if
it has a $\sigma$-finite carrier and is separable with respect to the topology
of local convergence in measure. We also address the question when a normed
lattice is a normal space with respect to the un-topology.
