Separation Axioms and Connectedness in Bounded Uniform Filter Spaces

Abstract

The well-known concepts such as uniform convergence, Cauchy convergence, Cartesian closed ness, quotient reflectiveness, quotients are not productive, boundedness, etc., are missing from Top, the category of topological spaces and continuous mappings. Mathematicians have taken several routes toward defining these ideas in Topology. In 2018, Dieter Leseberg introduced bounded uniform filter spaces that generalize topological spaces, bounded spaces, filter conver gence spaces, semiuniform convergence spaces, and bornological spaces based on the concept of filters of a set. This generalization overcomes almost all of the known deficiencies that ap peared in Top. The most basic reason for introducing bounded uniform filter spaces was to bring boundedness, convergence theories, uniformity, and topological concepts under a single umbrella. Bounded uniform filter spaces can be further subdivided into several isomorphic sub categories corresponding to various concepts in topological spaces. The dissertation is divided into six parts. In the first chapter, we review a few elementary categorical concepts. We recall some useful categorical spaces including preuniform convergence spaces, semiuniform convergence spaces, boundedness, bornological spaces, and bounded uniform filter (b-UFIL) spaces along with their corresponding morphisms respectively. The category of bounded uniform filter (b-UFIL) spaces and bounded uniformly continuous (buc) mappings, b-UFIL, is also proven to be a topological category. The second chapter provides a characterization of each of local T0 and local T1 objects in the category of b-UFIL spaces and examines their mutual relations. It is shown that every local T1 b-UFIL space is equivalent to a local T0 b-UFIL space, but the converse implication is not always true. We illustrate this with a few examples. In chapter three, we characterize each of T0 and T1 objects in the categories of several types of b UFIL spaces and examine their mutual relations. Moreover, we investigate the productivity and hereditary properties of T0 (resp. T1) bounded uniform filter spaces, and compare our findings to the usual T0 and T1 spaces. We show that every T0 (resp. T1) bounded uniform filter space satisfies the usual T0 (resp. the usual T1), although the converse is not always true. It is further proved that if a bounded uniform filter space is T0 at p and T1 at p for any p ∈ Z, then the space is T0 (resp. T1) overall. Finally, it is shown that the categories T0b-UFIL, T0b-UFIL and T1b-UFIL satisfy several properties such as epireflective and quotient-reflective subcategories of b-UFIL. Further, it is proved that T ′ 0 b-UFIL is a normalized, cartesian closed, and hereditary topological construct. This is summarized in a diagram. In chapter four, we characterize both closed and strongly closed subobjects in the category of b-UFIL spaces. We define two notions of closure operators and prove that they are (weakly) hereditary, idempotent, and productive closure operators of b-UFIL. Also, we introduce four different closure operators in bounded uniform filter spaces namely b-UFIL0cl, b-UFIL0scl, b-UFIL1cl and b-UFIL1scl. Using these closure operators we further characterize each of Tj (j = 0,1) b-UFIL spaces and examine that each of them forms quotient-reflective subcategories of b-UFIL. Moreover, these implications are summarized in a diagram. In the fifth chapter, we illustrate the concepts of connected and strongly connected bounded uni form filter spaces and prove that every strongly connected bounded uniform filter space is a con nected bounded uniform filter space, but the inverse implication is not always true. Finally, we introduce ultraconnected objects in a topological category and examine the relationship among connected, irreducible, and ultraconnected b-UFIL spaces. This is further summarized in a diagram. The findings from chapters two, three, four, and five are summed up in chapter six, along with some unresolved proposals and topics for future study.