Hostname: page-component-cb9f654ff-5jtmz Total loading time: 0 Render date: 2025-08-26T23:44:39.441Z Has data issue: false hasContentIssue false
  • English
  • Français

A NOTE ON RELATIVE PSEUDOCOMPACTNESS IN THE CATEGORY OF FRAMES

Part of: Boolean algebras (Boolean rings) Distributive lattices Maps and general types of spaces defined by maps

Published online by Cambridge University Press:  22 May 2012

THEMBA DUBE
THEMBA DUBE*
Affiliation:
Department of Mathematical Sciences, University of South Africa, PO Box 392, 0003 Pretoria, South Africa (email: dubeta@unisa.ac.za)
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

A subspace S of Tychonoff space X is relatively pseudocompact in X if every f∈C(X) is bounded on S. As is well known, this property is characterisable in terms of the functor υ which reflects Tychonoff spaces onto the realcompact ones. A device which exists in the category CRegFrm of completely regular frames which has no counterpart in Tych is the functor which coreflects completely regular frames onto the Lindelöf ones. In this paper we use this functor to characterise relative pseudocompactness.

Keywords

frameframe homomorphismreal-valued continuous functions on a frameLindelöf coreflectionrelative pseudocompactness

MSC classification

Secondary: 06D22: Frames, locales 54C40: Algebraic properties of function spaces

Information

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 87 , Issue 1 , February 2013 , pp. 120 - 130
Copyright
Copyright © Australian Mathematical Publishing Association Inc. 2012

References

[1]Baboolal, D. and Banaschewski, B., ‘Compactification and local connectedness of frames’, J. Pure Appl. Algebra 70 (1991), 3–16.CrossRefGoogle Scholar
[2]Ball, R. N. and Walters-Wayland, J., ‘C- and C *-quotients in pointfree topology’, Dissertationes Math. (Rozprawy Mat.) 412 (2002), 62.Google Scholar
[3]Banaschewski, B., The Real Numbers in Pointfree Topology, Textos de Matemática Série B, 12 (Departamento de Matemática da Universidade de Coimbra, Coimbra, 1997).Google Scholar
[4]Banaschewski, B. and Gilmour, C., ‘Stone–Čech compactification and dimension theory for regular σ-frames’, J. Lond. Math. Soc. 2(127) (1989), 1–8.CrossRefGoogle Scholar
[5]Banaschewski, B. and Gilmour, C., ‘Pseudocompactness and the cozero part of a frame’, Comment. Math. Univ. Carolin. 37(3) (1996), 577–587.Google Scholar
[6]Banaschewski, B. and Gilmour, C., ‘Realcompactness and the cozero part of a frame’, Appl. Categ. Structures 9 (2001), 395–417.CrossRefGoogle Scholar
[7]Banaschewski, B. and Mulvey, C., ‘Stone–Čech compactification of locales. I’, Houston J. Math. 6 (1980), 301–312.Google Scholar
[8]Blair, R. L. and Swardson, M. A., ‘Spaces with an Oz Stone–Čech compactification’, Topology Appl. 36 (1990), 73–92.CrossRefGoogle Scholar
[9]Dube, T., ‘Some notes on C- and C *-quotients of frames’, Order 25 (2008), 369–375.CrossRefGoogle Scholar
[10]Dube, T. and Matutu, P., ‘A few points on pointfree pseudocompactness’, Quaest. Math. 30 (2007), 451–464.CrossRefGoogle Scholar
[11]Dube, T. and Naidoo, I., ‘Round squares in the category of frames’, Houston J. Math., to appear.Google Scholar
[12]Dube, T. and Walters-Wayland, J., ‘Coz-onto frame maps and some applications’, Appl. Categ. Structures 15 (2007), 119–133.CrossRefGoogle Scholar
[13]Gillman, L. and Jerison, M., Rings of Continuous Functions (Van Nostrand, Princeton, 1960).CrossRefGoogle Scholar
[14]Madden, J. and Vermeer, J., ‘Lindelöf locales and realcompactness’, Math. Proc. Cambridge Philos. Soc. 99 (1986), 473–480.CrossRefGoogle Scholar
[15]Marcus, N., ‘Realcompactification of frames’, Comment. Math. Univ. Carolin. 36(2) (1995), 347–356.Google Scholar
[16]Picado, J. and Pultr, A., ‘Frames and locales: topology without points’, in: Frontiers in Mathematics (Birkhäuser, Basel, 2011).Google Scholar