\(T_ 2\)-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.}.


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


Zbl 0656.18010
Full Text: EuDML


[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
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.