Hostname: page-component-78c5997874-mlc7c Total loading time: 0 Render date: 2024-11-19T07:31:14.618Z Has data issue: false hasContentIssue false
  • English
  • Français

Fibrewise separation axioms for locales

Published online by Cambridge University Press:  24 October 2008

P. T. Johnstone
P. T. Johnstone
Affiliation:
Department of Pure Mathematics, University of Cambridge, 16 Mill Lane, Cambridge CB2 1SB
Get access Rights & Permissions [Opens in a new window]

Abstract

In this paper we study the weak versions of the fibrewise separation axioms for locales over a base locale, whose introduction has been made possible by the development, in a previous paper by the author, of the fibrewise notion of closure for sublocales of locales over a base. We establish the implications which hold between these axioms and the traditional separation axioms for locales, and give counter-examples to show that some of these implications are irreversible.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 108 , Issue 2 , September 1990 , pp. 247 - 256
Copyright
Copyright © Cambridge Philosophical Society 1990

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

REFERENCES

[1]Isbell, J. R.. Atomless parts of spaces. Math. Scand. 31 (1972), 5–32.CrossRefGoogle Scholar
[2]Isbell, J. R.. Function spaces and adjoints. Math. Scand. 36 (1975), 317–339.CrossRefGoogle Scholar
[3]Isbell, J. R.. Product spaces in locales. Proc. Amer. Math. Soc. 81 (1981), 116–118.Google Scholar
[4]Isbell, J. R.. Review of [12]. Bull. Amer. Math. Soc. (N.S.) 11 (1984), 389–392.Google Scholar
[5]Isbell, J. R.. First steps in descriptive theory of locales. Preprint, August 1989.Google Scholar
[6]James, I. M.. General Topology and Homotopy Theory (Springer-Verlag, 1984).Google Scholar
[7]James, I. M.. Spaces. Bull. Land. Math. Soc. 18 (1986), 529–559.CrossRefGoogle Scholar
[8]James, I. M.. Fibrewise Topology (Cambridge University Press, 1989).Google Scholar
[9]Jibladze, M. and Johnstone, P. T.. The frame of fibrewise closed nuclei. Cahiers Topologie Géom. Différentielle Catégoriques. (Submitted.)Google Scholar
[10]Johnstone, P. T.. The Gleason cover of a topos, II. J. Pure Appl. Alg. 22 (1981), 229–247.Google Scholar
[11]Johnstone, P. T.. The point of pointless topology. Bull. Amer. Math. Soc. (N.S.) 8 (1983), 41–53.CrossRefGoogle Scholar
[12]Johnstone, P. T.. Stone Spaces. Cambridge Studies in Advanced Math. no. 3 (Cambridge University Press, 1983).Google Scholar
[13]Johnstone, P. T.. Open locales and exponentiation. In Mathematical Applications of Category Theory, Contemp. Math. vol. 30 (American Mathematical Society, 1984), pp. 84–116.Google Scholar
[14]Johnstone, P. T.. A constructive ‘closed subgroup theorem’ for localic groups and groupoids. Cahiers Topologie Géom. Différentielle Catégoriques 30 (1989), 3–23.Google Scholar
[15]Johnstone, P. T.. A constructive theory of uniform locales. (Submitted.)Google Scholar
[16]Murchiston, G. S. and Stanley, M. G.. A ‘T 1’ space with no closed points, and a “T 1” locale which is not ‘T 1’. Math. Proc. Cambridge Philos. Soc. 95 (1984), 421–422.CrossRefGoogle Scholar
[17]Niefield, S. B.. Cartesian spaces over T and locales over Ω(T). Cahiers Topologie Géom. Différentielle Catégoriques 23 (1982), 257–267.Google Scholar
[18]Pultr, A.. Pointless uniformities, I: complete regularity. Comment. Math. Univ. Carolin. 25 (1984), 91–104.Google Scholar
[19]Simmons, H.. Spaces with Boolean assemblies. Colloq. Math. 43 (1980), 23–29.Google Scholar