Abstract
An analogue of Katětov's theorem on the equality between the dimension of a Tychonov space and the analytic dimension of its ring of bounded real-valued continuous maps is established for proximity spaces and proximally continuous maps by an internal method of proof. A new kind of filter, called proximally prime filter, arises naturally as a tool in this theory.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Čech, E.: Topological Spaces, Interscience, London, 1966.
Efremovič, V. A.: The geometry of proximity. I., Mat. Sbornik N. S. 31(73) (1952), 189–200.
Gillman, L. and Jerison, M.: Rings of Continuous Functions, Van Nostrand, Princeton, 1960.
Hejcman, J.. On analytical dimension of rings of bounded uniformly continuous functions, Comment. Math. Univ. Carol. 28 (1987), 325–335.
Herrlich, H.: Topologie II: Uniforme Räume, Heldermann-Verlag, Berlin, 1988. 041
Isbell, J. R.: Uniform Spaces, American Mathematical Society, Providence, 1964.
Katětov, M.: On rings of continuous functions and the dimension of compact spaces, Časopis Pěst Mat. Fys. 75 (1950), 1–16, (Russian, English and Czech summaries.)
Naimpally, S. A. and Warrack, B. D.: Proximity Spaces, Cambridge University Press, Cambridge, 1970.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Bentley, H.L., Hušek, M. & Ori, R.G. The Katětov dimension of proximity spaces. Appl Categor Struct 4, 43–55 (1996). https://doi.org/10.1007/BF00124113
Received:
Accepted:
Issue Date:
DOI: https://doi.org/10.1007/BF00124113