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

normal;pseudocomplement;disjunction;filter;Stone space;Boolean algebra

Seaton, Gerald Arthur ; Dr. Gary Faulkner, Committee Chair,Dr. Richard Chandler, Committee Member,Dr. Kailash Misra, Committee Member,Dr. Ernest Stitzinger, Committee Member,Seaton, Gerald Arthur ; Dr. Gary Faulkner ; Committee Chair ; Dr. Richard Chandler ; Committee Member ; Dr. Kailash Misra ; Committee Member ; Dr. Ernest Stitzinger ; Committee Member

University:North Carolina State University

关键词: normal;    pseudocomplement;    disjunction;    filter;    Stone space;    Boolean algebra;   

Others  :  https://repository.lib.ncsu.edu/bitstream/handle/1840.16/4123/etd.pdf?sequence=1&isAllowed=y

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【 摘 要 】

βXX 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 βXX.We subsequently give some examples.Most notably, for X = H this now provides a lattice-theoretic approach for representing βHH.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).