Pseudocompactness and the cozero part of a frame

Bernhard Banaschewski; Christopher Gilmour

Commentationes Mathematicae Universitatis Carolinae (1996)

  • Volume: 37, Issue: 3, page 577-587
  • ISSN: 0010-2628

Abstract

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A characterization of the cozero elements of a frame, without reference to the reals, is given and is used to obtain a characterization of pseudocompactness also independent of the reals. Applications are made to the congruence frame of a σ -frame and to Alexandroff spaces.

How to cite

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Banaschewski, Bernhard, and Gilmour, Christopher. "Pseudocompactness and the cozero part of a frame." Commentationes Mathematicae Universitatis Carolinae 37.3 (1996): 577-587. <http://eudml.org/doc/247922>.

@article{Banaschewski1996,
abstract = {A characterization of the cozero elements of a frame, without reference to the reals, is given and is used to obtain a characterization of pseudocompactness also independent of the reals. Applications are made to the congruence frame of a $\sigma $-frame and to Alexandroff spaces.},
author = {Banaschewski, Bernhard, Gilmour, Christopher},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {pseudocompact frames; $\sigma $-frames; cozero elements and Alexandroff spaces; pseudocompact frames; -frames; cozero elements and Alexandroff spaces},
language = {eng},
number = {3},
pages = {577-587},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {Pseudocompactness and the cozero part of a frame},
url = {http://eudml.org/doc/247922},
volume = {37},
year = {1996},
}

TY - JOUR
AU - Banaschewski, Bernhard
AU - Gilmour, Christopher
TI - Pseudocompactness and the cozero part of a frame
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 1996
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 37
IS - 3
SP - 577
EP - 587
AB - A characterization of the cozero elements of a frame, without reference to the reals, is given and is used to obtain a characterization of pseudocompactness also independent of the reals. Applications are made to the congruence frame of a $\sigma $-frame and to Alexandroff spaces.
LA - eng
KW - pseudocompact frames; $\sigma $-frames; cozero elements and Alexandroff spaces; pseudocompact frames; -frames; cozero elements and Alexandroff spaces
UR - http://eudml.org/doc/247922
ER -

References

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  1. Baboolal D., Banaschewski B., Compactification and local connectedness of frames, J. Pure Appl. Algebra 70 (1991), 3-16. (1991) Zbl0722.54031MR1100502
  2. Banaschewski B., The frame envelope of a σ -frame, Quaestiones Math. 16.1 (1993), 51-60. (1993) Zbl0779.06009MR1217474
  3. Banaschewski B., Frith J., Gilmour C., On the congruence lattice of a frame, Pacific J. Math. 130.2 (1987), 209-213. (1987) Zbl0637.06006MR0914098
  4. Banaschewski B., Gilmour C., Stone-Čech compactification and dimension theory for regular σ -frames, J. London Math. Soc. (2) No.127, 39, part 1 (1989), 1-8. Zbl0675.06005MR0989914
  5. Banaschewski B., Mulvey C., Stone-Čech compactification of locales I, Houston J. of Math. 6.3 (1980), 301-312. (1980) Zbl0473.54026MR0597771
  6. Banaschewski B., Mulvey C., Stone-Čech compactification of locales II, J. Pure Appl. Algebra 33 (1984), 107-122. (1984) Zbl0549.54017MR0754950
  7. Banaschewski B., Pultr A., Paracompactness revisited, Applied Categorical Structures 1 (1993), 181-190. (1993) Zbl0797.54032MR1245799
  8. Gilmour C., Realcompact Alexandroff spaces and regular σ -frames, PhD Thesis, University of Cape Town, 1981. Zbl0601.54019
  9. Gilmour C., Realcompact Alexandroff spaces and regular σ -frames, Math. Proc. Cambridge Philos. Soc. 96 (1984), 73-79. (1984) MR0743702
  10. Gordon H., Rings of functions determined by zero-sets, Pacific J. Math. 36 (1971), 133-157. (1971) Zbl0185.38803MR0320996
  11. Johnstone P.T., Stone Spaces, Cambridge Studies in Advanced Math. 3, Cambridge Univ. Press, 1982. Zbl0586.54001MR0698074
  12. Kennison J., m -Pseudocompactness, Trans. Amer. Math. Soc. 104 (1962), 436-442. (1962) Zbl0111.35004MR0145478
  13. Madden J., κ -Frames, J. Pure Appl. Algebra 70 (1991), 107-127. (1991) Zbl0721.06006MR1100510
  14. Madden J., Vermeer H., Lindelöf locales and realcompactness, Math. Proc. Camb. Phil. Soc. 99 (1986), 473-480. (1986) Zbl0603.54021MR0830360
  15. Marcus N., Realcompactifications of frames, MSc Thesis, University of Cape Town, 1994. 
  16. Reynolds G., On the spectrum of a real representable ring, Applications of Sheaves, Springer LNM 753 (1977), 595-611. Zbl0426.18002MR0555563
  17. Reynolds G., Alexandroff algebras and complete regularity, Proc. Amer. Math. Soc. 76 (1979), 322-326. (1979) Zbl0416.54015MR0537098
  18. Walters J., Uniform sigma frames and the cozero part of uniform frames, MSc Thesis, University of Cape Town, 1990. 
  19. Walters J., Compactifications and uniformities on sigma frames, Comment. Math. Univ. Carolinae 32.1 (1991), 189-198. (1991) Zbl0735.54014MR1118301

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