Semester

Summer

Date of Graduation

2020

Document Type

Dissertation

Degree Type

PhD

College

Eberly College of Arts and Sciences

Department

Mathematics

Committee Chair

Edgar Fuller

Committee Member

Harvey Diamond

Committee Member

Jerzy Wojciechowski

Committee Member

Adam Halasz

Committee Member

Ela Celikbas

Abstract

n this dissertation, we present new set functions called strongly limit and strongly prolongation limit sets. We show the new sets, especially strongly prolongation limit sets, characterize proper action under an arbitrary setting. That is, we characterize proper action for wider class of proper ܩ-spaces. Also, we show the new version of the sets could be derived from strongly exceptional sets which have been used as a good technique for the characterization of a proper maps. Moreover, we review properties of well-known limit sets and prolongations and properties for the new version of limit sets under an arbitrary setting on a ܩ-space ܺ. Next, we give characterizations for a proper action and a locally proper action using the new set functions and some relevant areas where one could use these sets to describe some topological properties for a space ܺ or algebraic properties for a group ܩ acting on ܺ. Following the technique of limit sets, we present a new version of action as weaker forms of the proper action called ݓ -actions using convergence techniques. This approach yields new directions for the proof of some existing theorems which are then given as well as come counter-examples. We give some applications for our weaker forms and how convergence techniques could be fruitful to explain the relations among certain notions in the theory of ܩ-spaces. Finally, according to our weaker forms, we characterize the notion of enough slice that has presented by H. Biller using ݓ -Cartan action.

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