Completeness properties of function rings in pointfree topology

Bernhard Banaschewski; Sung Sa Hong

Commentationes Mathematicae Universitatis Carolinae (2003)

  • Volume: 44, Issue: 2, page 245-259
  • ISSN: 0010-2628

Abstract

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This note establishes that the familiar internal characterizations of the Tychonoff spaces whose rings of continuous real-valued functions are complete, or σ -complete, as lattice ordered rings already hold in the larger setting of pointfree topology. In addition, we prove the corresponding results for rings of integer-valued functions.

How to cite

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Banaschewski, Bernhard, and Hong, Sung Sa. "Completeness properties of function rings in pointfree topology." Commentationes Mathematicae Universitatis Carolinae 44.2 (2003): 245-259. <http://eudml.org/doc/249183>.

@article{Banaschewski2003,
abstract = {This note establishes that the familiar internal characterizations of the Tychonoff spaces whose rings of continuous real-valued functions are complete, or $\sigma $-complete, as lattice ordered rings already hold in the larger setting of pointfree topology. In addition, we prove the corresponding results for rings of integer-valued functions.},
author = {Banaschewski, Bernhard, Hong, Sung Sa},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {frame of reals; lattice ordered rings of real valued continuous functions and integer valued continuous functions; extremally disconnected frame; basically disconnected frame; cozero map; frames; function rings},
language = {eng},
number = {2},
pages = {245-259},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {Completeness properties of function rings in pointfree topology},
url = {http://eudml.org/doc/249183},
volume = {44},
year = {2003},
}

TY - JOUR
AU - Banaschewski, Bernhard
AU - Hong, Sung Sa
TI - Completeness properties of function rings in pointfree topology
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 2003
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 44
IS - 2
SP - 245
EP - 259
AB - This note establishes that the familiar internal characterizations of the Tychonoff spaces whose rings of continuous real-valued functions are complete, or $\sigma $-complete, as lattice ordered rings already hold in the larger setting of pointfree topology. In addition, we prove the corresponding results for rings of integer-valued functions.
LA - eng
KW - frame of reals; lattice ordered rings of real valued continuous functions and integer valued continuous functions; extremally disconnected frame; basically disconnected frame; cozero map; frames; function rings
UR - http://eudml.org/doc/249183
ER -

References

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  1. Ball R.N., Walters-Wayland J.L., C - and C * -embedded sublocales, Dissertationes Math. 412 (2002). (2002) Zbl1012.54025MR1952051
  2. Banaschewski B., Universal zero-dimensional compactifications, In Categorical Topology and its Relation to Analysis, Algebra, and Combinatorics, Prague, Czechoslovakia, August 1988, World Scientific, Singapore, 1989, pp. 257-269. MR1047906
  3. Banaschewski B., The frame envelope of a σ -frame, Quaest. Math. 16 (1993), 51-60. (1993) Zbl0779.06009MR1217474
  4. Banaschewski B., The real numbers in pointfree topology, Textos de Matemática Série B, No. 12, Departamento de Matemática da Universidade de Coimbra, 1997. Zbl0891.54009MR1621835
  5. Banaschewski B., Mulvey C.J., 10.1016/0022-4049(84)90001-X, J. Pure Appl. Algebra 33 (1984), 107-122. (1984) Zbl0549.54017MR0754950DOI10.1016/0022-4049(84)90001-X
  6. Gillman L., Jerison M., Rings of continuous functions, D. Van Nostrand, 1960. Zbl0327.46040MR0116199
  7. Johnstone P.T., Conditions related to De Morgan's law, In Applications of Sheaves, Proceedings, Durham, 1977, Springer LNM 753 (1979), pp. 479-491. Zbl0445.03041MR0555556
  8. Johnstone P.T., Stone Spaces, Cambridge Studies in Advanced Mathematics 3, Cambridge University Press, 1982, . MR0698074
  9. Nakano H., Über das System aller stetigen Funktionen auf einen topologischen Raum, Proc. Imp. Acad. Tokyo 17 (1941), 308-310. (1941) MR0014173
  10. Stone M.H., 10.4153/CJM-1949-016-5, Canad. J. Math. 1 (1949), 176-186. (1949) Zbl0032.16901MR0029091DOI10.4153/CJM-1949-016-5
  11. Vickers S., Topology via Logic, Cambridge Tracts in Theor. Comp. Sci. No 5, Cambridge University Press, 1985. Zbl0922.54002MR1002193

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