# Article

 Title: Compactifications and uniformities on sigma frames  (English) Author: Walters, Joanne Language: English Journal: Commentationes Mathematicae Universitatis Carolinae ISSN: 0010-2628 (print) ISSN: 1213-7243 (online) Volume: 32 Issue: 1 Year: 1991 Pages: 189-198 . Category: math . Summary: A bijective correspondence between strong inclusions and compactifications in the setting of $\sigma$-frames is presented. The category of uniform $\sigma$-frames is defined and a description of the Samuel compactification is given. It is shown that the Samuel compactification of a uniform frame is completely determined by the $\sigma$-frame consisting of its uniform cozero part, and consequently, any compactification of any frame is so determined. Keyword: strong inclusion Keyword: compactification Keyword: uniform $\sigma$-frame Keyword: uniform cozero MSC: 18B35 MSC: 54D35 MSC: 54E05 MSC: 54E52 MSC: 54J05 idZBL: Zbl 0735.54014 idMR: MR1118301 . Date available: 2008-10-09T13:11:44Z Last updated: 2012-04-30 Stable URL: http://hdl.handle.net/10338.dmlcz/116954 . Reference: [1] Banaschewski B.: Frames and compactifications.Proc. I. International Symp. on Extension Theory of Topological Structures and Its Applications. VEB Deutscher Verlag der Wissenschaften, 1969. Zbl 0188.28006 Reference: [2] Banaschewski B., Gilmour C.R.A.: Stone-Čech compactification and dimension theory for regular $\sigma$-frames.J. London Math. Soc.(2) No. 127, 39, part 1 1-8 (1989). Zbl 0675.06005, MR 0989914 Reference: [3] Banaschewski B., Mulvey C.: Stone-Čech compactification of Locales I..Houston J. of Math. 6, 3 (1980), 301-312. Zbl 0473.54026, MR 0597771 Reference: [4] Frith J.L.: The Category of Uniform Frames.Cahier de Topologie et Geometrie, to appear. Zbl 0738.18002, MR 1109372 Reference: [5] Gilmour C.R.A.: Realcompact Alexandroff spaces and regular $\sigma$-frames.Math. Proc. Cambridge Philos. Soc. 96 (1984), 73-79. MR 0743702 Reference: [6] Ginsburg S., Isbell J.R.: Some operators on uniform spaces.Trans. Amer. Math. Soc. 36 (1959), 145-168. Zbl 0087.37601, MR 0112119 Reference: [7] Johnstone P.T.: Stone Spaces.Cambridge Studies in Advanced Math. 3, Cambridge Univ. Press, 1982. Zbl 0586.54001, MR 0698074 Reference: [8] Madden J., Vermeer H.: Lindelöf locales and realcompactness.Math. Proc. Camb. Phil. Soc. (1985), 1-8. Reference: [9] Pultr A.: Pointless uniformities I. Complete regularity.Comment. Math. Univ. Carolinae 25 (1984), 91-104. Zbl 0543.54023, MR 0749118 .

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