Use of generalizations of topologies and uniformities, by Dieter Leseberg

Topology Atlas Document # taic-17.htm | Production Editor: R. Flagg

Topology Atlas Invited Contributions, vol. 1, issue 5 (1996), 91.

Use of generalizations of topologies and uniformities
by

Dieter Leseberg

(Universitaetsbibliothek der Technischen Universitaet Braunschweig, Postfach 3329, 38023 Braunschweig, Germany)

Since many desirable topological and uniform statements, unfortunately, fail to be true, several of these can be made true provided the familiar constructions of subspaces, products and quotients are replaced by more appropiate constructions. So, the categories of topological or uniform spaces have the property that initial structures exist to arbitrary families of maps (in the sense of Bourbaki), but lack other desirable properties such as "cartesian closedness" or "extensionality", which means that natural function space structures exist or moreover that quotients are hereditary. A possible approach to improve the situation is to look at "extensions" which have (some of) these properties; of course the originally given category should be nicely embedded in such extensions.

In 1964 Doitchinov introduced the notion of supertopological spaces in order to construct a unified theory of topological spaces, proximity spaces and uniform spaces. In 1973 Herrlich introduced nearness spaces and related generalizations respectively and since that time these spaces have been used for different purpurses by topologists. In 1987 Tozzi and Wyler considered generalized proximity relations over B-sets and obtained a topological category which contains both the supertopological spaces and moreover the topogenous spaces in the sense of Csaszar, which are isomorphic to the quasi-proximities as defined by Fletcher and Lindgren. So, the supernearness spaces introduced by myself present a common unification of all the above concepts and moreover it is possible to characterize those supernearness spaces which can be extended to topological spaces. The so called grill-setting semi-supernearness spaces have natural function spaces which means it is possible to supply for any pair (X,Y) of grill-setting semi-supernearness spaces the set [X,Y] of all sn-maps from Y to X with a decent grill-setting semi-supernearness, e.g. such that [[X,Y],Z] is natural isomorphic to [X,Y+Z] or at least to [[X,Z],Y]. Because of this fact, which is inconvenient for investigations in algebraic topology (homotopy theory), functional analysis (duality theory) or topological algebra (quotients), Top has been substituted by well-behaved and more convenient supercategories!

Some nice reviews with respect to the above considered problems:

H. Herrlich, Categorical Topology 1971-1981 in: Proc. Fith Prague Topological Symposium 1981 (Heldermann, Berlin 1983), 279-383.

H. Herrlich, Topological Structures in: Math. Centre Tracts 52 (Amsterdam 1974), 59-122.

D.C.Kent and Nandita Rath, Filter spaces. Appl. Categorical Structures 1 (1993), 297-309.

W.Gaehler, A topological approach to structure theory. Math. Nachr. 100 (1981), 93-144.

G. Preuss, Semiuniform Convergence Spaces. Prep. Nr.A-24-94 (FU Berlin, FB Mathematik und Informatik), 1-32.

A. Csaszar, m-contiguities I-III. Lecture given at the 1994 Amsterdam Conference and it is submitted for publication to Acta Mathematica Hungaria.

H.L.Bentley, H.Herrlich and E.Lowen-Colebunders, Convergence. Journal of Pure and Applied Algebra 68 (1990), 27-45.

Received by the editors: January 17, 1996.