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Nontransitive quasi-uniformities
by
Hans-Peter A. Künzi
University of Cape Town
In this talk a quasi-uniformity is a uniformity that does not necessarily satisfy the symmetry condition. A quasi-uniformity is called transitive provided that it has a base consisting of transitive entourages.
From the beginning a basic problem in the theory of quasi-uniform spaces was the question whether certain quasi-uniformities are transitive. The following results are typical.
(1) The fine quasi-uniformity of the Kofner plane is not transitive.
(2) The fine quasi-uniformity of each suborderable (Kofner), each orthocompact semistratifiable (Junnila) and each T1-space with an ortho-base (Kofner) is transitive.
(3) There exists a nontransitive functorial quasi-uniformity coarser than the locally finite covering quasi-uniformity on Top (Brümmer, Künzi).
In our talk we shall deal with the following more recent contributions to our subject.
(A) The fine quasi-uniformity of each hereditarily metacompact locally compact regular space is transitive (Künzi, Junnila, Watson).
(B) If a topological space admits at least two quasi-uniformities then it admits at least 2c nontransitive as well as at least 2c transitive quasi-uniformities (Künzi, Losonczi).
(C) Each infinite completely regular (T2-)space admits at least 2c nontransitive as well as at least 2c transitive totally bounded quasi-uniformities; for each nonzero cardinal \kappa there is a topological space admitting exactly \kappa totally bounded quasi-uniformities, all of which are transitive (Gerlits, Künzi, Losonczi, Pérez-Peñalver, Szentmiklóssy).
We shall also shortly discuss the following questions that are still open.
(I) Is the fine quasi-uniformity of each (quasi-metrizable) Moore space (Junnila) or each non-archimedeanly quasi-metrizable space (Fletcher, Lindgren) transitive?
(II) In ZFC, is there a compact Hausdorff space whose finest quasi-uniformity is not transitive (van Douwen)?
(III) For an arbitrary topological space, does a quasi-proximity class containing a non-totally-bounded member possess (at least 2c) nontransitive members (Künzi, Losonczi)?
Reference: H.-P. A. Künzi, Nonsymmetric distances and their associated topologies: About the origins of basic ideas in the area of asymmetric topology, in: Handbook of the History of General Topology, Vol. 3, eds. C. Aull and R. Lowen, Kluwer, Dordrecht, 2001 (to appear).
Date received: February 15, 2001
Copyright © 2001 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # cagx-02.