PROPERTIES OF ZERO-ONE VALUED MEASURES AND THEIR APPLICATION TO TOPOLOGY.

摘要

This work continues the work started by Cohen (9) and Koltun (14). They worked with zero-one valued measures defined boolean algebra generated by a lattice L. A connection was established between the properties of zero-one measures on L and topological properties of L.; Here the connection is further strengthened. Chapter One looks at some old and new lattice topological properties and relates them to measures. Two of the properties we look at are almost compact, and almost countably compact. We define a new type of measure property, we denote {dollar}Phi{dollar}(L). We relate this new class of measures to complement generated lattices. In Chapter Two we take {dollar}Phi{dollar}(L) and topologize it with its associated lattice. We define semi-prime complete and relate it to other forms of completeness. We also look at the associated lattice of measures concentrated at points. We end the chapter defining a new measure property we call weakly regular. In Chapter Three we look at two types of zero-one valued outer measures. We relate the outer measures to weakly regular measures, and use them to establish some new results. We end the chapter defining a co-complement generated lattice and relate it to measure and topological properties of the lattice. We introduce the concept of weakly countably compact in terms of measures. In Chapter Four we study a one-to-one, onto operator we call an anti-isomorphism. We use it to prove that if all zero-one valued measures on a lattice are regular then the lattice is complemented. In Chapter Five we end our work applying our previous results to a point set framework. We show that the lattice of closed sets almost countably compact implies the set is pseudo-compact. We define regular countably compact and show this implies almost countably compact. We generalize regular countably compact to regular compact and show this is equivalent to lattice of regular closed sets being compact. We show in the point set framework what almost compact looks like, and also investigate the properties of C-compact, semi-normal, almost regular, and semi-regular. We end by giving representations of pseudocompactness.