Hostname: page-component-78c5997874-94fs2 Total loading time: 0 Render date: 2024-11-12T20:31:56.530Z Has data issue: false hasContentIssue false
  • English
  • Français

Addendum to ‘The Katětov construction modified for a T0-quasi-metric space’

Published online by Cambridge University Press:  12 November 2014

HANS-PETER A. KÜNZI  and
MANUEL SANCHIS
HANS-PETER A. KÜNZI
Affiliation:
Department of Mathematics and Applied Mathematics, University of Cape Town, Rondebosch 7701, South Africa Email: [email protected]
MANUEL SANCHIS
Affiliation:
Institut Universitari de Matemàtiques i Aplicacions (IMAC), Universitat Jaume I de Castelló, Spain Email: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

It is known that if K is a compact subset of the (separable complete) metric Urysohn space (${\mathbb U}$, d) and f is a Katětov function on the subspace K of (${\mathbb U}$, d), then there is z ∈ ${\mathbb U}$ such that d(z, x) = f(x) for all x ∈ K.

Answering a question of Normann, we show in this article that the supseparable bicomplete q-universal ultrahomogeneous T0-quasi-metric space (q${\mathbb U}$, D) recently discussed by the authors satisfies a similar property for Katětov function pairs on subsets that are compact in the associated metric space (q${\mathbb U}$, Ds).

Type
Paper
Information
Mathematical Structures in Computer Science , Volume 25 , Special Issue 8: Computing with Infinite Data: Topological and Logical Foundations Part 2 , December 2015 , pp. 1685 - 1691
Copyright
Copyright © Cambridge University Press 2014 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Dress, A. W. M. (1984) Trees, tight extensions of metric spaces, and the cohomological dimension of certain groups: A note on combinatorial properties of metric spaces. Advances in Mathematics 53 321–402.Google Scholar
Fletcher, P. and Lindgren, W. F. (1982) Quasi-Uniform Spaces, Dekker, New York.Google Scholar
Gromov, M. (1998) Metric Structures for Riemannian and Non-Riemannian Spaces, Birkhäuser, Basel 78–85.Google Scholar
Huhunaišvili, G. E. (1955) On a property of Urysohn's universal metric space. Doklady Akademii Nauk SSSR (Russian) 101 607–610.Google Scholar
Kamo, H. (2005) Computability and computable uniqueness of Urysohn's universal metric space. In: Grubba, T., Hertling, P., Tsuiki, H. and Weihrauch, K. (eds.) CCA2005 149–159.Google Scholar
Katětov, M. (1988) On universal metric spaces, in general topology and its relations to modern analysis and algebra VI. In: Frolík, Z., (ed.) Proceedings of the 6th Prague Topological Symposium, 1986, Heldermann Verlag, Berlin 323–330.Google Scholar
Kemajou, E., Künzi, H.-P. A. and Otafudu, O. O. (2012) The Isbell-hull of a di-space. Topology Application 159 2463–2475.CrossRefGoogle Scholar
Künzi, H.-P. A. (2009) An introduction to quasi-uniform spaces. In: Mynard, F. and Pearl, E. (eds.) Beyond Topology, Contemporary mathematics volume 486 239–304.Google Scholar
Künzi, H.-P. A. and Sanchis, M. (2012) The Katětov construction modified for a T 0-quasi-metric space. Topology Application 159 711–720.Google Scholar
Lešnik, D. (2008) Constructive Urysohn's universal metric space. Electronic Notes in Theoretical Computer Science 221 171–179.Google Scholar
Lešnik, D. (2009) Constructive Urysohn universal metric space. Journal of Universal Computer Science 15 (6) 1236–1263.Google Scholar
Melleray, J. (2007) On the geometry of Urysohn's universal metric space. Topology Application 154 384–403.Google Scholar
Normann, D. (2009) A rich hierarchy of functionals of finite types. Logical Methods in Computer Science 5 (3:11) 1–21.Google Scholar
Urysohn, P. (1927) Sur un espace métrique universel. Bulletin des Sciences Mathematiques 51 43–64, 74–90.Google Scholar