Hostname: page-component-cd9895bd7-jkksz Total loading time: 0 Render date: 2024-12-26T12:53:33.320Z Has data issue: false hasContentIssue false
  • English
  • Français

T1-locales

Published online by Cambridge University Press:  24 October 2008

J. Rosický  and
B. Šmarda
J. Rosický
Affiliation:
Department of Mathematics, Purkynĕ University, 66295 Brno, Czechoslovakia
B. Šmarda
Affiliation:
Department of Mathematics, Purkynĕ University, 66295 Brno, Czechoslovakia
Get access Rights & Permissions [Opens in a new window]

Extract

In the theory of locales (or ‘pointless topology'), several authors have tried to find a suitable form of a T1-separation axiom. Their proposals are unsatisfactory in the sense that they do not coincide for topological spaces with the T1-axiom. Our main result is that sublocales of sober T1-spaces are exactly locales in which primes are dual atoms. Hence, these locales should be called T1, despite the fact that they include any locale without points. Our T1-locales are also closed with respect to products; in fact they form the smallest epireflective subcategory of locales containing all sober T1-spaces. Besides T1-locales we will also consider T2-locales and epireflective sub-categories of locales in general.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 98 , Issue 1 , July 1985 , pp. 81 - 86
Copyright
Copyright © Cambridge Philosophical Society 1985

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]Banaschewski, B. and Harting, R.. Lattice aspects of radical ideals and choice principles. Proc. London Math. Soc. (3) 50 (1985), 385–404.CrossRefGoogle Scholar
[2]Dowker, C. H. and Strauss, D.. Separation axioms for frames. Coll. Math. Soc. J. Bolyai 8 (1974), 223–240.Google Scholar
[3]Dowker, C. H. and Strauss, D.. T1 and T2-axioms for frames. In Aspects of Topology: In Memory of Hugh Dowker, L.M.S. Lecture Notes Series no. 93 (Cambridge University Press, 1985), 325–335.CrossRefGoogle Scholar
[4]Fourman, M. P.. T1-spaces over topological sites. J. Pure Appl. Algebra 27 (1983), 223–224CrossRefGoogle Scholar
[5]Herrlich, H.. On the concept of reflection in general topology. Proc. Symp. Extension Theory of Topol. Structures. (VEB Deutsch. Verl. d. Wissenschaften, 1969), 105–114.Google Scholar
[6]Herrlich, H.. Topologische Reflexionen und Coreflexionen, Lect. Notes in Math. 78 (Springer- Verlag, 1968).CrossRefGoogle Scholar
[7]Herrlich, H. and Strecker, G. E.. Category Theory (Allyn and Bacon, 1973).Google Scholar
[8]Hofmann, K. H. and Lawson, J. D.. The spectral theory of distributive continuous lattices. Trans. Amer. Math. Soc. 246 (1978), 285–310.CrossRefGoogle Scholar
[9]Isbell, J. R.. Atomless parts of spaces. Math. Scand. 31 (1972), 5–32.CrossRefGoogle Scholar
[10]Isbell, J. R.. Function spaces and adjoints. Math. Scand. 36 (1975), 317–339.CrossRefGoogle Scholar
[11]Johnstone, P. T.. Stone spaces. (Cambridge University Press, 1982).Google Scholar
[12]Kerstan, J.. Verallgemeinerung eines Satzes von Tarski. Math. Nachr. 17 (1958-1959), 16–18.CrossRefGoogle Scholar
[13]Murchiston, G. S. and Stanley, M. G.. A ‘T1’ space with no closed points and a "T1" locale which is not ‘T1’. Math. Proc. Cambridge Philos. Soc. 95 (1984), 421–422.CrossRefGoogle Scholar
[14]Simmons, H.. The lattice theoretic part of topological separation properties. Proc. Edinburgh Math. Soc. (2) 21 (1978), 41–48.CrossRefGoogle Scholar