Order theoretic and topological aspects of generalizations of well ordered sets

Abstract

Set theory, order theory and topology are closely interconnected disciplines that provide a framework for the study of mathematical structures. The interactions among these areas have proved instrumental in a wide range of applications, including computer algorithms, formal languages and automata theory, combinatorics, mathematical modelling, algebra, logic and related fields. Motivated by the study of anti-homogeneous topological spaces, the notion of semi-well ordered sets was introduced to characterize such spaces. This thesis undertakes a detailed investigation of the set-theoretic and order-theoretic properties of semi-well ordered sets. Illustrative examples are presented, additional structural propertics are examined and order ideals are studied. Furthermore, the concept of semi-ordinals is introduced and their arithmetic and fundamental properties are explored. We further study the order topology on semi-well ordered sets and show that these spaces are zero-dimensional and strongly locally compact. The isolated points of semi-well ordered sets are identified and characterizations of infinite homogeneous and compact semi-well ordered sets are obtained. In addition, continuous functions between semi-well ordered sets are characterized and it is shown that any function between two such sets is the point-wise limit of a net of continuous functions. Further topological properties are examined and the dynamics of semi-ordinals are also investigated. We introduce partially semi-well ordered sets as a generalization of both semi-well ordered sets and partially well ordered sets. The main motivation for this generalization is to verify the Aharoni-Korman conjecture (k = 1) for this broader class of partially ordered sets. We investigate the order-theoretic and structural properties of partially semi-well ordered sets, the properties of their order ideals and their reversibility. Using a specific structural characterization, we verify the Aharoni-Korman conjecture (k = 1) for partially semi-well ordered sets. The non-existence of an infinite antichain in a partially semi-well ordered set ensures that the interval topology and the Dedekind topology coincide on these sets; we refer to this common topology as the intrinsic topology. Conditions for Dedekind completeness, compactness and connectedness of this space are examined. We characterize the closure of subsets and establish a relationship between the relative topology and the intrinsic topology on subsets of partially semi-well ordered sets. We further show that this topological space is anti-rigid and we also discuss additional properties of this topological space. Finally, we characterize continuous functions between partially semi-well ordered sets and derive a result concerning functions between such sets based on this characterization.