  A topological space $X$ is called resolvable if it contains a dense subset
with dense complement. Using only basic principles, we show that whenever the
space $X$ has a resolving subset that can be written as an at most countably
infinite union of subsets, in such a way that a given vector lattice of (not
necessarily bounded) continuous functions on $X$ separates every point outside
the resolving subset from each of its constituents, then the order continuous
dual of this lattice is trivial. In order to apply this result in specific
cases, we show that several spaces have resolving subsets that can be written
as at most countably infinite unions of closed nowhere dense subsets. An appeal
to the main result then yields, for example, that, under appropriate
conditions, vector lattices of continuous functions on separable spaces, metric
spaces, and topological vector spaces have trivial order continuous duals if
they separate points and closed nowhere dense subsets. Our results in this
direction extend known results in the literature. We also show that, under
reasonably mild separation conditions, vector lattices of continuous functions
on locally connected $T_1$ Baire spaces without isolated points have trivial
order continuous duals. A discussion of the relation between our results and
the non-existence of non-zero normal measures is included.
