| Title:
|
Totally bounded frame quasi-uniformities (English) |
| Author:
|
Fletcher, P. |
| Author:
|
Hunsaker, W. |
| Author:
|
Lindgren, W. |
| Language:
|
English |
| Journal:
|
Commentationes Mathematicae Universitatis Carolinae |
| ISSN:
|
0010-2628 (print) |
| ISSN:
|
1213-7243 (online) |
| Volume:
|
34 |
| Issue:
|
3 |
| Year:
|
1993 |
| Pages:
|
529-537 |
| . |
| Category:
|
math |
| . |
| Summary:
|
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 $(L, {\bold U})$ are meaningful for quasi-uniform frames. If ${\bold U}$ is a totally bounded quasi-uniformity on a frame $L$, there is a totally bounded quasi-uniformity $\overline{{\bold U}}$ on $\Re L$ such that $(\Re L, \overline{{\bold U}})$ is a compactification of $(L,{\bold U})$. Moreover, the Cauchy spectrum of the uniform frame $(Fr({\bold U}^{\ast }), {\bold U}^{\ast })$ can be viewed as the spectrum of the bicompletion of $(L,{\bold U})$. (English) |
| Keyword:
|
frame |
| Keyword:
|
uniform frame |
| Keyword:
|
quasi-uniform frame |
| Keyword:
|
quasi-proximity |
| Keyword:
|
totally bounded quasi-uniformity |
| Keyword:
|
uniformly regular ideal |
| Keyword:
|
compactification |
| Keyword:
|
bicompletion |
| MSC:
|
06D20 |
| MSC:
|
18B35 |
| MSC:
|
54D35 |
| MSC:
|
54E05 |
| MSC:
|
54E15 |
| idZBL:
|
Zbl 0786.54028 |
| idMR:
|
MR1243084 |
| . |
| Date available:
|
2009-01-08T18:05:53Z |
| Last updated:
|
2012-04-30 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/118609 |
| . |
| Reference:
|
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| Reference:
|
[2] Banaschewski B., Brümmer G.C.L., Hardie K.A.: Biframes and bispaces.Quaestiones Math. 6 (1983), 13-25. MR 0700237 |
| Reference:
|
[3] Banaschewski B., Pultr A.: Samuel compactification and completion of uniform frames.Math. Proc. Camb. Phil. Soc. (1) 108 (1990), 63-78. Zbl 0733.54020, MR 1049760 |
| Reference:
|
[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. Zbl 0114.14101, MR 0146792 |
| Reference:
|
[5] Fletcher P., Hunsaker W.: Entourage uniformities for frames.Monatsh. Math. 112 (1991), 271-279. Zbl 0736.54023, MR 1141095 |
| Reference:
|
[6] Fletcher P., Hunsaker W.: Symmetry conditions in terms of open sets.Topology and its Appl. 45 (1992), 39-47. Zbl 0766.54025, MR 1169075 |
| Reference:
|
[7] Fletcher P., Hunsaker W., Lindgren W.: Characterizations of Frame Quasi-Uniformities.preprint. Zbl 0792.54026 |
| Reference:
|
[8] Fletcher P., Hunsaker W., Lindgren W.: Frame Quasi-Uniformities.preprint. Zbl 0796.54037 |
| Reference:
|
[9] Fletcher P., Lindgren W.: Quasi-Uniform Spaces.Marcel Dekker, New York and Basel, 1982. Zbl 0583.54017, MR 0660063 |
| Reference:
|
[10] Frith J.L.: Structured Frames.Ph.D. Thesis, Univ. Cape Town, 1987. |
| Reference:
|
[11] Gantner T.E., Steinlage R.C.: Characterizations of quasi-uniformities.J. London Math. Soc. (2) 5 (1972), 48-52. Zbl 0241.54023, MR 0380741 |
| Reference:
|
[12] Hunsaker W., Lindgren W.F.: Construction of quasi-uniformities.Math. Ann. 188 (1970), 39-42. Zbl 0187.44602, MR 0266149 |
| Reference:
|
[13] Isbell J.R.: Uniform Spaces.Mathematical Surveys, No. 12, Amer. Math. Soc., Providence, R.I., 1964. Zbl 0124.15601, MR 0170323 |
| Reference:
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[14] Johnstone P.T.: Stone Spaces.Cambridge Univ. Press, Cambridge, 1982. Zbl 0586.54001, MR 0698074 |
| Reference:
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[15] Nachbin L.: Sur les espaces uniformes ordonnés.C.R. Acad. Sci., Paris 226 (1948), 774-775. Zbl 0030.37303, MR 0024120 |
| Reference:
|
[16] Pultr A.: Pointless uniformities I. Complete regularity.Comment Math. Univ. Carolinae 25 (1984), 91-104. Zbl 0543.54023, MR 0749118 |
| . |
