Semisolid sets and topological measures

This paper is one in a series that investigates topological measures on locally compact spaces. A topological measure is defined on open and closed sets; it is finitely additive on the collection of open and compact sets, inner regular on open sets, and outer regular on closed sets. We examine semisolid sets and give a way of constructing topological measures from solid-set functions on locally compact, Hausdorff, connected, locally connected spaces. For compact spaces we present two simpler methods than the currently available one. We give examples of finite and infinite topological measures on locally compact spaces and present an easy way to generate topological measures on spaces whose one-point compactification has genus 0. Our results are necessary for various methods for constructing topological measures, give additional properties of topological measures, and provide a tool for determining whether two topological measures or quasi-linear functionals are the same.