Protopological groups and other generalizations of topological groups.

摘要

A protopological group is a group G with a topology tau such that there exists a collection N of normal subgroups satisfying (1) For every neighborhood U of the identity, there exists an N ∈ N such that N ⊆ U and (2) G/N with the quotient topology is a topological group for every N ∈ N . The collection N is called a normal system. The collection of quotient topologies is called a quotient system. Protopological groups were first studied by Judith Covington in 1993. The purpose of this dissertation is to extend Covington's work and to study a broader class of groups with topologies from a categorical point of view.;In Chapter Two, we give a Characterization Theorem, which completely describes all topologies on a group that have the same normal system and the same quotient system. We use this to show that the product of protopological groups is a protopological group. We also show how homomorphisms between groups can be used to generate protopological topologies.;In Chapter Three, we study separation and topological properties, such as compactness and connectedness. We give examples to show that there are properties that topological groups possess, but protopological groups do not possess. However, we also show that many results, particularly those concerning open and closed subgroups, concerning topological groups hold in the category of protopological groups.;Since a protopological group is a generalization of a topological group, in Chapter Four we address the question of when a protopological group is a topological group. In particular, we investigate topological properties and properties of the normal system.;In Chapter Five, we show that the theory of protopological groups can be used to study a much broader class of groups. We also generalize the idea of a protopological group and define a pro-P group. We give a Characterization Theorem for Pro-P Groups, which is analogous to the Characterization Theorem in Chapter Two. We also show that some results concerning protopological groups hold in the category of pro-P groups.