$$T_ 2$$-frames and almost compact frames.(English)Zbl 0779.54015

The authors introduce a “Hausdorff axiom” for locales which is equivalent to that introduced, from a slightly different point of view, by the reviewer and Sun Shu-Hao [Weak products and Hausdorff locales, in “Categorical algebra and its applications”, Proc. 1st Conf., Louvain-la-Neuve/Belg. 1987, Lect. Notes Math. 1348, 173-193 (1988; Zbl 0656.18010)], and investigate its basic properties; they also consider weak compactness and the localic version of the Katětov $$H$$- closed extension. {Although this paper was written some five years before it was published, the standard of proof-reading of the published version leaves a good deal to be desired.}.

MSC:

 54D10 Lower separation axioms ($$T_0$$–$$T_3$$, etc.) 54D35 Extensions of spaces (compactifications, supercompactifications, completions, etc.) 06D99 Distributive lattices

Zbl 0656.18010
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References:

 [1] B. Banaschewski and R. Harting: Lattice aspects of radical ideals and choice principles. Proc. London Math. Soc. (3) 50 (1985), 384-404. · Zbl 0569.16003 [2] B. Banaschewski and C. J. Mulvey: Stone-Čech compactification of locales I. Houston J. Math. 6 (1980), 301-312. · Zbl 0473.54026 [3] A. Czászár: General topology. Akademiai Kiado, Budapest, 1978. [4] E. Čech: Topological spaces. Academia, Praha, 1966. · Zbl 0141.39401 [5] C. H. Dowker and D. Strauss: Separation axioms for frames. Coll. Math. Soc. Janos Bolyai 8 (1974), 223-240. · Zbl 0293.54001 [6] C. H. Dowker and D. Strauss: $$T_1$$- and $$T_2$$-axioms for frames. Aspects of Topology: In Memory of Hugh Dowker, L. M. S. Lecture Notes Series No. 93, Cambridge University Press, 1985, pp. 325-335. [7] C. H. Dowker and D. Strauss: Sums in the category of frames. Houston J. Math. 3 (1976), 17-32. · Zbl 0341.54001 [8] H. Herrlich: Topologische Reflexionen und Coreflexionen. Lect. Notes in Math. 78, Springer-Verlag, 1968. · Zbl 0182.25302 [9] J. R. Isbell: Atomless parts of spaces. Math. Scand. 31 (1972), 5-32. · Zbl 0246.54028 [10] P. T. Johnstone: Stone spaces. Cambridge University Press, 1982. · Zbl 0499.54001 [11] P. T. Johnstone and Sun Shu-Hao: Weak products and Hausdorff locales. preprint. · Zbl 0656.18010 [12] J. I. Kerstan: Verallgemeinerung eines Satzes von Tarski. Math. Nachr. 17 (1958-9), 16-18. · Zbl 0081.38501 [13] I. Kříž: A constructive proof of the Tychonoff’s theorem for locales. Comm. Math, Univ. Carolinae 26, 3 (1985), 619-630. · Zbl 0661.54027 [14] G. S. Murchiston and M. G. Stanley: A ”$$T_1$$” space with no closed points and a ”$$T_1$$” locale which is not ”$$T_1$$”. Math. Proc. Cambridge Philos. Soc. 85 (1984), 421-422. · Zbl 0537.54001 [15] A. Pultr: Some recent results of the theory of locales. Sixth Prague Topological Symposium, 1986. [16] J. Rosický and B. Šmarda: $$T_1$$-locales. Math. Proc. Cambridge Philos. Soc. 98 (1985), 81-86. · Zbl 0596.54019 [17] H. Simmons: The lattice theoretic part of topological separation properties. Proc. Edinburgh Math. Soc. (2) 21 (1978), 41-48. · Zbl 0396.54014
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