On order continuous duals of vector lattices of continuous functions on resolvable spaces

TL;DR

This paper establishes conditions under which the order continuous duals of vector lattices of continuous functions on certain resolvable spaces are trivial, extending existing results in the theory of vector lattices and topology.

Contribution

It introduces a new criterion involving resolvable subsets for the triviality of order continuous duals of vector lattices of continuous functions.

Findings

Spaces with resolvable subsets as countable unions of closed nowhere dense sets have trivial duals.

Separable, metric, and topological vector spaces with point-separating lattices have trivial duals under certain conditions.

Locally connected T1 Baire spaces without isolated points also have trivial duals.

Abstract

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.