IJPAM: Volume 60, No. 4 (2010)
Invited Lecture Delivered at
Fifth International Conference of Applied Mathematics
and Computing (Plovdiv, Bulgaria, August 1218, 2008)

TOPOLOGICAL EXTENSIONS IN
THE REALM OF BCONVERGENCE
Department of Mathematics and Informatics
Free University of Berlin
45, Habelschwerdter Allee, Berlin, 14195, GERMANY
email: [email protected]
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 Hausdorffextensions of a given topological space are
closely related with a class of supertopologies on , which he
called bsupertopologies. 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
grilldetermined, more precisely, is the subset of some bunch. The
concepts mentioned above both are subsumed by the socalled
supernearness spaces introduced by myself in 2002. Consequently,
their corresponding grilldefined 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 setconvergences introduced by Wyler in 1988 as well as
grilldefined supernear operators cannot be subsumed by PUCONV.
This motivated establishing our broader concept of bconvergence
(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, setconvergence, nearness spaces, supernear operators, preuniform convergence, bconvergence, topological extensions
Source: International Journal of Pure and Applied Mathematics
ISSN: 13118080
Year: 2010
Volume: 60
Issue: 4