Every finite system of T1 uniformities comes from a single distance structure
DOI:
https://doi.org/10.4995/agt.2002.2113Keywords:
Distance function, Free monoid, Generalized metric, UniformityAbstract
Using the general notion of distance function introduced in an earlier paper, a construction of the finest distance structure which induces a given quasi-uniformity is given. Moreover, when the usual defining condition xy : d(y; x) of the basic entourages is generalized to nd(y; x) n (for a fixed positive integer n), it turns out that if the value-monoid of the distance function is commutative, one gets a countably infinite family of quasi-uniformities on the underlying set. It is then shown that at least every finite system and every descending sequence of T1 quasi-uniformities which fulfil a weak symmetry condition is included in such a family. This is only possible since, in contrast to real metric spaces, the distance function need not be symmetric.
Downloads
References
M. M. Bonsangue, F. van Breugel, and J. J. M. M. Rutten, Generalized metric spaces: completion, topology, and powerdomains via the Yoneda embedding, Theoret. Comput. Sci. 193 (1998), no. 1-2, 1-51.
Maurice Fréchet, Sur les classes V normales, Trans. Amer. Math. Soc. 14 (1913), 320-325. http://dx.doi.org/10.2307/1988600
Jobst Heitzig, Partially ordered monoids and distance functions, Diploma thesis, Universität Hannover, Germany, July 1998.
Jobst Heitzig, Many familiar categories can be interpreted as categories of generalized metric spaces, Appl. Categ. Structures (to appear) (2002). http://dx.doi.org/10.1023/A:1020526912025
Ralph Kopperman, All topologies come from generalized metrics, Amer. Math. Monthly 95 (1988), no. 2, 89-97.
Djuro R. Kurepa, General ecart, Zb. Rad. (1992), no. 6, part 2, 373-379.
Boyu Li, Wang Shangzi, and Maurice Pouzet, Topologies and ordered semigroups, Topology Proc. 12 (1987), 309-325.
Karl Menger, Untersuchungen über allgemeine Metrik, Math. Annalen 100 (1928), 75-163. http://dx.doi.org/10.1007/BF01448840
J. Nagata, A survey of the theory of generalized metric spaces, (1972), 321-331.
Maurice Pouzet and Ivo Rosenberg, General metrics and contracting operations, Discrete Math. 130 (1994), 103-169. http://dx.doi.org/10.1016/0012-365X(92)00527-X
Hans-Christian Reichel, Distance-functions and g-functions as a unifying concept in the theory of generalized metric spaces, Recent developments of general topology and its applications (Berlin, 1992), Akademie-Verlag, Berlin, 1992, p. 279-286.
B. Schweizer and A. Sklar, Probabilistic metric spaces, North-Holland, New York, 1983.
Downloads
Published
How to Cite
Issue
Section
License
This journal is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike- 4.0 International License.