Publication Date

5-29-2012

Advisor(s) - Committee Chair

Dr. Tom Richmond, Director, Dr. Ferhan Atici, Dr. John Spraker

Degree Program

Department of Mathematics and Computer Science

Degree Type

Master of Science

Abstract

The aim of this thesis is to establish the principal properties for the theory of ordered compactifications relating to connectedness and to provide particular examples. The initial idea of this subject is based on the notion of the Stone-Cech compactification.
The ordered Stone-Cech compactification oX of an ordered topological space X is constructed analogously to the Stone-Cech compactification X of a topological space X, and has similar properties. This technique requires a conceptual understanding of the Stone-Cech compactification and how its product applies to the construction of ordered topological spaces with continuous increasing functions. Chapter 1 introduces background information.

Chapter 2 addresses connectedness and compactification. If (A;B) is a separation of
a topological space X, then (A 8 B) = A 8 B, but in the ordered setting, o(A 8 B)
need not be oA 8 oB. We give an additional hypothesis on the separation (A;B) to
make o(A 8 B) = oA 8 oB. An open question in topology is when is X -X = X. We
answer the analogous question for ordered compactifications of totally ordered spaces. So, we are concerned with the remainder, that is, the set of added points oX -X. We
demonstrate the topological properties by using lters. Moreover, results of lattice theory turn out to be some of the basic tools in our original approach.

In Chapter 3, specific examples and counterexamples are given to illustrate earlier
results.

Disciplines

Mathematics

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