Nonsymmetric distance structures

Hans-Peter A. Künzi

If we omit the symmetry condition in the usual definition of an entourage uniformity we obtain the concept of a (filter) quasi-uniformity. Similarly, if we omit the symmetry condition in the definition of a (pseudo)metric, we reach the notion of a quasi-(pseudo)metric. Many classical counterexamples in topology like the Sorgenfrey plane, the Niemytzki plane, the Michael line or the Pixley-Roy space over the reals are (obviously) quasi-metrizable.

Asymmetric distance functions had already been considered by Hausdorff in the beginning of the 20^th century when in his classical book on set-theory he discussed the Hausdorff metric of a metric space. Later they were dealt with by Niemytzki when he explored the interplay of the various assumptions in the usual axiomatization of a metric space.

The study of quasi-uniformities began in 1948 with Nachbin's investigations on uniform preordered spaces, that is, those topological preordered spaces whose preorder is given by the intersection of the entourages of a filter quasi-uniformity U and whose topology is induced by the associated sup-uniformity U v U^-1. He showed that the topological ordered spaces of this kind can be characterized by the property that they admit T₂-order compactifications. The term quasi-proximity first appeared in the articles of Pervin and Steiner. The connection between (totally bounded) quasi-uniformities and quasi-proximities is analogous to the symmetric case.

The filter U^-1 of inverse relations of a quasi-uniformity U is also a quasi-uniformity. Similarly each quasi-pseudometric has an obvious conjugate by interchanging the order of points. Hence quasi-uniformities and quasi-metrics naturally generate bitopological spaces (in the sense of Kelly), that is, sets equipped with two topologies. Császár developed the theory of the bicompletion for quasi-uniform structures that was later popularized by Fletcher, Lindgren and Salbany. Subsequently, Doitchinov found a completion theory for his so-called quiet quasi-uniformities.

Brümmer was first to consider explicitly the class of all functorial quasi-uniformities, although some basic work on canonical covering quasi-uniformities was done at about the same time by Fletcher and Lindgren.

The work of Fox, Junnila and Kofner showed that the classes of \gamma-spaces (= T₁-spaces admitting a local quasi-uniformity with a countable base), quasi-metrizable spaces and non-archimedeanly quasi-metrizable spaces are all distinct and that the fine quasi-uniformity of metrizable and suborderable (= generalized ordered) spaces has a base consisting of transitive entourages.

Recently, the interest in quasi-uniform function spaces and hyperspaces increased considerably. Partially this fact can be explained by their application in computer science. In theoretical computer science several authors tried to develop with the help of quasi-pseudometrics a common generalization of the two well-established theories of metric spaces and partial orders that would contain the classical results as special cases. In connection with such investigations also the old idea to replace the reals in the definition of a quasi-pseudometric by some more general structure was investigated again (e.g. in the studies on Kopperman's continuity spaces).

Covering quasi-uniformities in the pointless context were first studied by Frith in his thesis. In the last years Fletcher, Hunsaker, Lindgren and Picado developed a theory of entourage quasi-uniformities for frames. Many authors tried more or less successfully to extend major classical results about quasi-uniformities to some fuzzy setting. Based on Lowen's theory of approach spaces, Windels introduced the concept of an approach quasi-uniformity.

Quasi-uniform structures were also investigated in various kinds of topological algebraic structures. In particular the study of paratopological groups with the help of quasi-uniformities is well known. (A paratopological group is a group equipped with a topology such that the group operation is continuous.) Recently further applications of methods from the theory of quasi-uniform spaces to problems in approximation theory and functional analysis were discovered. On the other hand, over the years, numerous tools from various mathematical theories like lattice theory, category theory, nonstandard analysis and descriptive set-theory were applied successfully to the study of quasi-uniform structures.

Some well-known open problems

1. Is each quasi-metrizable Moore space non-archimedeanly quasi-metrizable?
2. Does the finest compatible quasi-uniformity of a non-archimedeanly quasi-metrizable space possess a transitive base (= a base consisting of transitive entourages)?
3. If a quasi-proximity class contains more than one quasi-uniformity, does it necessarily contain a nontransitive member (that is, a member without transitive base)?
4. Does each infinite Hausdorff space admit a nontransitive totally bounded quasi-uniformity?
5. Is there in ZFC a compact Hausdorff space whose finest compatible quasi-uniformity does not have a transitive base?
6. Is there a countably orthocompact \sigma-orthocompact (regular) topological space that is not orthocompact?

Monographs

[A] P. Fletcher and W.F. Lindgren, Quasi-Uniform Spaces, Lecture Notes Pure Appl. Math. 77, Dekker, New York, 1982.

[B] E.M. Jawhari, D. Misane and M. Pouzet, Retracts: Graphs and ordered sets from the metric point of view, Contemp. Math. 57 (1986), 175-226.

[C] K. Keimel and W. Roth, Ordered Cones and Approximation, Lecture Notes Math. 1517, Springer, Berlin, 1992.

[D] R. Lowen, Approach Spaces: The Missing Link in the Topology - Uniformity - Metric Triad, Oxford Mathematical Monographs, Oxford University Press, 1997.

[E] M.G. Murdeshwar and S.A. Naimpally, Quasi-Uniform Topological Spaces, Nordhoff, Groningen, 1966.

[F] E.M. Zaustinsky, Spaces with non-symmetric distance, Memoirs Amer. Math. Soc. 34, 1959.

Survey Articles

[1] G.C.L. Brümmer, Categorical aspects of the theory of quasi-uniform spaces, Rend. Ist. Mat. Univ. Trieste 30 Suppl. (1999), 45-74.

[2] A. Császár, Old and new results on quasi-uniform extension, Rend. Ist. Mat. Univ. Trieste 30 Suppl. (1999), 75-85.

[3] J. Deák, A survey of compatible extensions (presenting 77 unsolved problems), Colloq. Math. Soc. János Bolyai 55, Top., Pécs (Hungary), 1989, (1993), 127-175.

[4] J. Kofner, On quasi-metrizability, Topology Proc. 5 (1980), 111-138.

[5] R.D. Kopperman, Asymmetry and duality in topology, Topology Appl. 66 (1995), 1-39.

[6] H.P. A. Künzi, Quasi-uniform spaces - eleven years later, Topology Proc. 18 (1993), 143-171.

[7] H.P. A. Künzi, Nonsymmetric topology, Bolyai Society in Mathematical Studies 4, Topology, Szekszárd, 1993, Hungary, (Budapest 1995), 303-338.

[8] H.P. A. Künzi, Quasi-uniform isomorphism groups, in Recent Progress in Function Spaces, eds. G. Di Maio and L. Holá, Quaderni di matematica, Dip. Mat. Sec. Univ. Napoli, Vol. 3 (1998), 187-220.

[9] H.P. A. Künzi, Nonsymmetric distances and their associated topologies: About the origins of basic ideas in the area of asymmetric topology, in C.E. Aull and R. Lowen, eds., Handbook of the History of General Topology, Vol. 3, Dordrecht, Kluwer (to appear).

[10] H.P. A. Künzi, Preservation of completeness under mappings in asymmetric topology, Applied General Topology 1 (2000), 117-134.

Some recent articles

[1] M.M. Bonsangue, J.J.M.M. Rutten and F. van Breugel, Generalized metric spaces: completion, topology, and powerdomains via the Yoneda embedding, Theoretical Computer Science 193 (1998), 1-51.

[2] J. Cao, I.L. Reilly and S. Romaguera, Some properties of quasiuniform multifunction spaces, J. Austral. Math. Soc. (Ser. A) 64 (1998), 169-177.

[3] K. Ciesielski, R.C. Flagg and R. Kopperman, Polish spaces, computer approximation and cocompact quasimetrizability, preprint.

[4] R.C. Flagg, R. Kopperman and P. Sünderhauf, Smyth completion as bicompletion, Topology Appl. 91 (1999), 169-180.

[5] J.L. Frith and W. Hunsaker, Completion of quasi-uniform frames, Appl. Categ. Struct. 7 (1999), 263-272.

[6] J. Gerlits, H.P. A. Künzi, A. Losonczi and Z. Szentmiklóssy, The existence of compatible nontransitive totally bounded quasi-uniformities, Topology Appl. (to appear).

[7] R. Heckmann, Approximation of metric spaces by partial metric spaces, Appl. Categ. Struct. 7 (1999), 71-83.

[8] M. Henriksen, R. Kopperman, J. Mack and D.W.B. Somerset, Joincompact spaces, continuous lattices and C*-algebras, Algebra Univ. 38 (1997), 289-323.

[9] H.P. A. Künzi and A. Losonczi, On some cardinal functions related to quasi-uniformities, Houston J. Math. 26 (2000), 299-313.

[10] H.P. A. Künzi, J. Marín and S. Romaguera, Quasi-uniformities on topological semigroups and bicompletion, Semigroup Forum (to appear).

[11] H.P. A. Künzi and S. Romaguera, Well-quasi-ordering and the Hausdorff quasi-uniformity, Topology Appl. 85 (1998), 207-218.

[12] M. Nauwelaerts, The Cartesian closed topological hull of the category of (quasi)uniform spaces, preprint.

Date published: July 29, 2000.