IJPAM: Volume 60, No. 4 (2010)

Invited Lecture Delivered at
Fifth International Conference of Applied Mathematics
and Computing (Plovdiv, Bulgaria, August 12-18, 2008)



TOPOLOGICAL EXTENSIONS IN
THE REALM OF B-CONVERGENCE

Dieter Leseberg
Department of Mathematics and Informatics
Free University of Berlin
45, Habelschwerdter Allee, Berlin, 14195, GERMANY
e-mail: d.leseberg@tu-bs.de


Abstract.In 1964 Doitchinov introduced the notion of supertopological spaces in order to construct a unified theory of topological, proximity and uniform spaces. As an application he showed that the compactly determined Hausdorff-extensions of a given topological space $X$ are closely related with a class of supertopologies on $X$, which he called b-supertopologies. In 1973 Herrlich introduced nearness spaces, which generalize both symmetrical topological spaces and uniform spaces. Bentley showed that those nearness spaces that can be extended to a topological space have an elegant internal characterization, namely that every nearness collection is grill-determined, more precisely, is the subset of some bunch. The concepts mentioned above both are subsumed by the so-called supernearness spaces introduced by myself in 2002. Consequently, their corresponding grill-defined supernear operators are describing these extensions in a common manner. The topological construct PUCONV of preuniform convergence spaces introduced by Preußin 1993 plays an important role in the study of strong topological universes, in which ``convergence structures'' are available, among them the filtermerotopies in the sense of Katetov, generalized convergence spaces, and uniform convergence structures such as quasiuniformities and various generalizations as well. Consequently, natural function spaces exist in such categories (i.e., they are Cartesian closed), quotients are stable under products, and in addition such categories are extensional. Here we only point out that supertopologies and their generalized set-convergences introduced by Wyler in 1988 as well as grill-defined supernear operators cannot be subsumed by PUCONV. This motivated establishing our broader concept of b-convergence (see IJPAM 2008). Within this concept we will now examine how topological extensions can be described.

Received: August 14, 2008

AMS Subject Classification: 54A20, 54B30, 54D35, 54E15, 54E17

Key Words and Phrases: supertopological spaces, set-convergence, nearness spaces, supernear operators, preuniform convergence, b-convergence, topological extensions

Source: International Journal of Pure and Applied Mathematics
ISSN: 1311-8080
Year: 2010
Volume: 60
Issue: 4