The lattice of equivalence classes of closed sets and the Stone-C?ech compactification

by Seaton, Gerald Arthur.

SEATON, GERALD ARTHUR. The Lattice of Equivalence Classes of Closed Sets and the Stone- · Cech Compacti¯cation. (Under the direction of Dr. Gary Faulkner.) ¯X n X is the remainder of the Stone- ·Cech compacti¯cation 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 ultra¯lters on L(X)) is homeomorphic to ¯X n X. We subsequently give some examples. Most notably, for X = H this now provides a lattice-theoretic approach for representing ¯H n 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 ultra¯lters on it. We also investigate some of the lattice properties of L(X).