Doctor of Philosophy (PhD), Ohio University, 2017, Mathematics (Arts and Sciences)

The field of Selection Principles in Mathematics is in some sense the study of diagonalization processes. It has its roots in a few basic selection principles that arose from the study of problems in analysis, dimension theory, topology, and set theory. These "classical" selection principles were formally defined by M. Scheepers in 1996, but they go back to classical works of F. Rothberger, W. Hurewicz, and K. Menger. Since then, new selection principles and new types of covers have been introduced and studied in relation to the classical selection principles. We consider the relationship between gamma-sets, which are spaces satisfying a specific classical selection principle, and a newer selection principle (A, B_∞) that was introduced by B. Tsaban in 2007. First, we survey known results of gamma-sets due to F. Galvin and A.W. Miller and prove which results hold for the (A, B_∞) selection principle. Then, we establish filter characterizations for these selection principles to prove new properties and positively answer a question asked by B. Tsaban. Afterward, we prove several results about a concrete construction of a gamma-set on the Cantor space due to T. Orenshtein and B. Tsaban. Lastly, we revisit our properties to formulate some open questions raised by our work.

Committee: Todd Eisworth (Advisor); Vladimir Uspenskiy (Committee Member); Sergio Lopez (Committee Member); Phillip Ehrlich (Committee Member) Subjects: Mathematics