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Keywords:
strong inclusion; compactification; uniform $\sigma$-frame; uniform cozero
strong inclusion; compactification; uniform $\sigma$-frame; uniform cozero
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.
References:
[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
[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). MR 0989914 | Zbl 0675.06005
[3] Banaschewski B., Mulvey C.: Stone-Čech compactification of Locales I. Houston J. of Math. 6, 3 (1980), 301-312. MR 0597771 | Zbl 0473.54026
[4] Frith J.L.: The Category of Uniform Frames. Cahier de Topologie et Geometrie, to appear. MR 1109372 | Zbl 0738.18002
[5] Gilmour C.R.A.: Realcompact Alexandroff spaces and regular $\sigma$-frames. Math. Proc. Cambridge Philos. Soc. 96 (1984), 73-79. MR 0743702
[6] Ginsburg S., Isbell J.R.: Some operators on uniform spaces. Trans. Amer. Math. Soc. 36 (1959), 145-168. MR 0112119 | Zbl 0087.37601
[7] Johnstone P.T.: Stone Spaces. Cambridge Studies in Advanced Math. 3, Cambridge Univ. Press, 1982. MR 0698074 | Zbl 0586.54001
[8] Madden J., Vermeer H.: Lindelöf locales and realcompactness. Math. Proc. Camb. Phil. Soc. (1985), 1-8.
[9] Pultr A.: Pointless uniformities I. Complete regularity. Comment. Math. Univ. Carolinae 25 (1984), 91-104. MR 0749118 | Zbl 0543.54023
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