Hostname: page-component-cd9895bd7-fscjk Total loading time: 0 Render date: 2024-12-27T03:31:11.318Z Has data issue: false hasContentIssue false
  • English
  • Français

Lindelöf locales and ℕ-compactness

Published online by Cambridge University Press:  24 October 2008

J. Paseka
J. Paseka
Affiliation:
J. E. Purkyně University, Brno, Czechoslovakia
Get access Rights & Permissions [Opens in a new window]

Extract

In the theory of locales (pointless topologies) some lattice-theoretical properties of certain classes of locales (Lindelöf, paracompact locales) behave ‘better’ than in the case of topological spaces. In this context we show that a locale is localic ‘ℕ-compact’ if and only if it is 0-dimensional Lindelöf. This is an analogue of the theorem of Madden and Vermeer [9] that a locale possesses the localic version of real compactness if and only if it is Tychonoff Lindelöf. As a consequence, we offer a localic analogue of ℕ-compactification, which is exactly the 0-dimensional Lindelöf reflection.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 109 , Issue 1 , January 1991 , pp. 187 - 191
Copyright
Copyright © Cambridge Philosophical Society 1991

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 Mulvey, C. J.. Stone—Čech compactification of locales. I. Houston J. Math. 6 (1980), 301–312.Google Scholar
[2]Dowker, C. H. and Strauss, D.. Sums in the category of frames. Houston J. Math. 3 (1976), 17–32.Google Scholar
[3]Engelking, R.. On functions defined on cartesian products, Fund. Math. 59 (1966), 221–231.CrossRefGoogle Scholar
[4]Engelking, R.. General Topology Polish Scientific Publishers, Warsaw (1977).Google Scholar
[5]Engelking, R. and Mrówka, S.. On E-compact spaces. Bull. Acad. Polon. Sci. Ser. Sci. Math. Astronom. Phys. 6 (1958), 429–436.Google Scholar
[6]Gilmour, C. R. A.. Realcompact spaces and regular σ-frames. Math. Proc. Cambridge Philos. Soc. 96 (1984), 73–79.CrossRefGoogle Scholar
[7]Herrlich, H.. E-kompakte Räume. Math. Z. 96 (1967), 228–255.CrossRefGoogle Scholar
[8]Johnstone, P. T.. Stone Spaces. Cambridge Studies in Advanced Math. no. 3 (Cambridge University Press, 1982).Google Scholar
[9]Madden, J. and Vermeer, J.. Lindelöf locales and realcompactness. Math. Proc. Cambridge Philos. Soc. 99 (1986), 473–480.CrossRefGoogle Scholar
[10]Reynolds, G.. On the spectrum of a real representable ring. In Applications of Sheaves. Lecture Notes in Math. vol. 753 (Springer-Verlag, 1979), pp. 595–611.CrossRefGoogle Scholar
[11]Rosický, J. and Šmarda, B.. T 1-locales. Math. Proc. Cambridge Philos. Soc. 96 (1985), 81–86.CrossRefGoogle Scholar