The Lattice of Equivalence Classes of Closed Sets and the Stone-Cech Compactification.

by Seaton, Gerald Arthur

SEATON, GERALD ARTHUR. The Lattice of Equivalence Classes of Closed Sets and the Stone-Cech Compactification. (Under the direction of Dr. Gary Faulkner.) ßX X is the remainder of the Stone-Cech compactification of a locally compact space X. This paper introduces a lattice which we call L(X) that is constructed using equivalence classes of closed sets of X. We then determine that St(L(X)) (the set of ultrafilters on L(X)) is homeomorphic to ßX X. We subsequently give some examples. Most notably, for X = H this now provides a lattice-theoretic approach for representing ßH H. In addition, we expand and clarify some aspects of lattice theory related to our constructions. We introduce the term "upwardly nonlinear" as a way to describe lattices with a certain property related to the ultrafilters on it. We also investigate some of the lattice properties of L(X).