Topic of Research Seminar: Separation Axioms and Connectedness in Bounded Uniform Filter Spaces

Abstract: It is well-known that Top, the category of topological spaces and continuous maps, fails to have the concept of uniform convergence, Cauchy convergence, Cartesian closedness, quotient reflective, the product of quotients, boundedness, etc. Several approaches have been made by mathematicians to define these concepts in the realm of topology. As a remedy to this defect that overcomes almost all the known deficiencies that appeared in Top, in 2018, Dieter Lese- berg introduced bounded uniform filter spaces which is a generalized structure based on the concept of filters of a set, and a generalization of topological spaces, filter convergence spaces, semiuni form convergence spaces, bounded spaces and bornological spaces. There were several motivations behind introducing bounded uniform convergence spaces and the most fundamental one was to unify convergence theories, boundedness, topological and uniformity concepts. There can be defined numerous isomorphic subcategories of the category of bounded uniform filter spaces correspond as different concepts in topological spaces.

This dissertation consists of six chapters. In the first chapter, some fundamental categorical concepts, various spaces such as preuniform convergence spaces, semiuniform convergence spaces, boundedness, bornological spaces, bounded uniform filter spaces, and their respective maps are recalled. Moreover, it is shown that b-UFIL, the category of bounded uniform filter spaces and bounded uniformly continuous maps, is a topological category. In the second chapter, local T0 and local T1 bounded uniform filter spaces are characterized, and their relation to each other is examined. In the third chapter, T0 and T1 bounded uniform filter spaces are characterized and their relation- ship with each other is examined. In addition, obtained results are compared with the usual T0 and T1 axioms, and hereditary and productivity properties of T0 (resp. T1) for bounded uniform filter spaces are determined. Furthermore, it is proved that every T0 (resp. T1) bounded uniform filter space implies the usual T0 (resp. the usual T1) but the inverse implication is not true in general. Moreover, quotient reflective subcategories of the category of bounded uniform filter spaces were determined.

In the fourth chapter, closed and strongly closed bounded uniform filter spaces are characterized, and their relations to each other are examined. Moreover, closure operators in bounded uniform filter spaces were examined and their relationship with separation axioms was discussed.

In the fifth chapter, connected and strongly connected bounded uniform filter spaces are developed. Also, irreducibility and ultraconnectedness in bounded uniform filter spaces were examined, and their relation with connectedness was discussed. In the last chapter, results obtained from chapters two, three, four, and five are summarized and several unsolved proposals and issues for further research are discussed.

Subject Field of Topic: Mathematics

Name of Speaker: Ms. Sana Khadim

Professorial Rank of Speaker: PhD Scholar (Mathematics Dpt.)

University Email of Speaker: [email protected]

Affiliation of Speaker: NUST School of Natural Sciences (NUST-SNS)

Date and Venue: October 19, 2022 at 1530 hrs, School of Natural Sciences (SN) Old building CR # 303, NUST Islamabad Campus