Totally bounded frame quasi-uniformities

Peter Fletcher; Worthen N. Hunsaker; William F. Lindgren

Commentationes Mathematicae Universitatis Carolinae (1993)

  • Volume: 34, Issue: 3, page 529-537
  • ISSN: 0010-2628

Abstract

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This paper considers totally bounded quasi-uniformities and quasi-proximities for frames and shows that for a given quasi-proximity ◃ on a frame L there is a totally bounded quasi-uniformity on L that is the coarsest quasi-uniformity, and the only totally bounded quasi-uniformity, that determines ◃ . The constructions due to B. Banaschewski and A. Pultr of the Cauchy spectrum ψ L and the compactification ℜ L of a uniform frame ( L , 𝐔 ) are meaningful for quasi-uniform frames. If 𝐔 is a totally bounded quasi-uniformity on a frame L , there is a totally bounded quasi-uniformity 𝐔 ¯ on ℜ L such that ( ℜ L , 𝐔 ¯ ) is a compactification of ( L , 𝐔 ) . Moreover, the Cauchy spectrum of the uniform frame ( F r ( 𝐔 * ) , 𝐔 * ) can be viewed as the spectrum of the bicompletion of ( L , 𝐔 ) .

How to cite

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Fletcher, Peter, Hunsaker, Worthen N., and Lindgren, William F.. "Totally bounded frame quasi-uniformities." Commentationes Mathematicae Universitatis Carolinae 34.3 (1993): 529-537. <http://eudml.org/doc/247453>.

@article{Fletcher1993,
abstract = {This paper considers totally bounded quasi-uniformities and quasi-proximities for frames and shows that for a given quasi-proximity $\triangleleft $ on a frame $L$ there is a totally bounded quasi-uniformity on $L$ that is the coarsest quasi-uniformity, and the only totally bounded quasi-uniformity, that determines $\triangleleft $. The constructions due to B. Banaschewski and A. Pultr of the Cauchy spectrum $\psi L$ and the compactification $\Re L$ of a uniform frame $(L, \{\mathbf \{U\}\})$ are meaningful for quasi-uniform frames. If $\{\mathbf \{U\}\}$ is a totally bounded quasi-uniformity on a frame $L$, there is a totally bounded quasi-uniformity $\overline\{\{\mathbf \{U\}\}\}$ on $\Re L$ such that $(\Re L, \overline\{\{\mathbf \{U\}\}\})$ is a compactification of $(L,\{\mathbf \{U\}\})$. Moreover, the Cauchy spectrum of the uniform frame $(Fr(\{\mathbf \{U\}\}^\{\ast \}), \{\mathbf \{U\}\}^\{\ast \})$ can be viewed as the spectrum of the bicompletion of $(L,\{\mathbf \{U\}\})$.},
author = {Fletcher, Peter, Hunsaker, Worthen N., Lindgren, William F.},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {frame; uniform frame; quasi-uniform frame; quasi-proximity; totally bounded quasi-uniformity; uniformly regular ideal; compactification; bicompletion; quasi-proximity; uniformly regular ideal; uniform frame; quasi-uniform frames; totally bounded quasi-uniformity; compactification; bicompletion},
language = {eng},
number = {3},
pages = {529-537},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {Totally bounded frame quasi-uniformities},
url = {http://eudml.org/doc/247453},
volume = {34},
year = {1993},
}

TY - JOUR
AU - Fletcher, Peter
AU - Hunsaker, Worthen N.
AU - Lindgren, William F.
TI - Totally bounded frame quasi-uniformities
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 1993
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 34
IS - 3
SP - 529
EP - 537
AB - This paper considers totally bounded quasi-uniformities and quasi-proximities for frames and shows that for a given quasi-proximity $\triangleleft $ on a frame $L$ there is a totally bounded quasi-uniformity on $L$ that is the coarsest quasi-uniformity, and the only totally bounded quasi-uniformity, that determines $\triangleleft $. The constructions due to B. Banaschewski and A. Pultr of the Cauchy spectrum $\psi L$ and the compactification $\Re L$ of a uniform frame $(L, {\mathbf {U}})$ are meaningful for quasi-uniform frames. If ${\mathbf {U}}$ is a totally bounded quasi-uniformity on a frame $L$, there is a totally bounded quasi-uniformity $\overline{{\mathbf {U}}}$ on $\Re L$ such that $(\Re L, \overline{{\mathbf {U}}})$ is a compactification of $(L,{\mathbf {U}})$. Moreover, the Cauchy spectrum of the uniform frame $(Fr({\mathbf {U}}^{\ast }), {\mathbf {U}}^{\ast })$ can be viewed as the spectrum of the bicompletion of $(L,{\mathbf {U}})$.
LA - eng
KW - frame; uniform frame; quasi-uniform frame; quasi-proximity; totally bounded quasi-uniformity; uniformly regular ideal; compactification; bicompletion; quasi-proximity; uniformly regular ideal; uniform frame; quasi-uniform frames; totally bounded quasi-uniformity; compactification; bicompletion
UR - http://eudml.org/doc/247453
ER -

References

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  2. Banaschewski B., Brümmer G.C.L., Hardie K.A., Biframes and bispaces, Quaestiones Math. 6 (1983), 13-25. (1983) MR0700237
  3. Banaschewski B., Pultr A., Samuel compactification and completion of uniform frames, Math. Proc. Camb. Phil. Soc. (1) 108 (1990), 63-78. (1990) Zbl0733.54020MR1049760
  4. Dowker C.H., Mappings of proximity structures, ``General Topology and its Relations to Modern Analysis and Algebra'', (Proc. Sympos. Prague 1961), Academic Press, New York, 1962, 139-141 Publ. House Czech. Acad. Sci., Prague, 1962. Zbl0114.14101MR0146792
  5. Fletcher P., Hunsaker W., Entourage uniformities for frames, Monatsh. Math. 112 (1991), 271-279. (1991) Zbl0736.54023MR1141095
  6. Fletcher P., Hunsaker W., Symmetry conditions in terms of open sets, Topology and its Appl. 45 (1992), 39-47. (1992) Zbl0766.54025MR1169075
  7. Fletcher P., Hunsaker W., Lindgren W., Characterizations of Frame Quasi-Uniformities, preprint. Zbl0792.54026
  8. Fletcher P., Hunsaker W., Lindgren W., Frame Quasi-Uniformities, preprint. Zbl0796.54037
  9. Fletcher P., Lindgren W., Quasi-Uniform Spaces, Marcel Dekker, New York and Basel, 1982. Zbl0583.54017MR0660063
  10. Frith J.L., Structured Frames, Ph.D. Thesis, Univ. Cape Town, 1987. 
  11. Gantner T.E., Steinlage R.C., Characterizations of quasi-uniformities, J. London Math. Soc. (2) 5 (1972), 48-52. (1972) Zbl0241.54023MR0380741
  12. Hunsaker W., Lindgren W.F., Construction of quasi-uniformities, Math. Ann. 188 (1970), 39-42. (1970) Zbl0187.44602MR0266149
  13. Isbell J.R., Uniform Spaces, Mathematical Surveys, No. 12, Amer. Math. Soc., Providence, R.I., 1964. Zbl0124.15601MR0170323
  14. Johnstone P.T., Stone Spaces, Cambridge Univ. Press, Cambridge, 1982. Zbl0586.54001MR0698074
  15. Nachbin L., Sur les espaces uniformes ordonnés, C.R. Acad. Sci., Paris 226 (1948), 774-775. (1948) Zbl0030.37303MR0024120
  16. Pultr A., Pointless uniformities I. Complete regularity, Comment Math. Univ. Carolinae 25 (1984), 91-104. (1984) Zbl0543.54023MR0749118

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