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Insertion, approximation, and extension of real-valued functions. (English) Zbl 0558.54012
Authors’ summary: ”For a uniformly closed vector lattice V of real-valued functions on a set X, necessary and sufficient conditions are obtained for insertion (or ”strict insertion”) of some member of V between two arbitrary real-valued functions on X. These conditions quickly yield known insertion, approximation, and extension theorems for real-valued functions.”
Reviewer: K.Kutzler

MSC:
54C30 Real-valued functions in general topology
54C20 Extension of maps
46A40 Ordered topological linear spaces, vector lattices
54C35 Function spaces in general topology
54C45 \(C\)- and \(C^*\)-embedding
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[1] R. Baire, Sur les séries à termes continus et tous de même signe, Bull. Soc. Math. France 32 (1904), 125 – 128 (French). · JFM 35.0271.01
[2] Robert L. Blair, Extensions of Lebesgue sets and of real-valued functions, Czechoslovak Math. J. 31(106) (1981), no. 1, 63 – 74. With a loose Russian summary. · Zbl 0481.54009
[3] Nicolas Bourbaki, Elements of mathematics. General topology. Part 1, Hermann, Paris; Addison-Wesley Publishing Co., Reading, Mass.-London-Don Mills, Ont., 1966. Nicolas Bourbaki, Elements of mathematics. General topology. Part 2, Hermann, Paris; Addison-Wesley Publishing Co., Reading, Mass.-London-Don Mills, Ont., 1966.
[4] Jean Dieudonné, Une généralisation des espaces compacts, J. Math. Pures Appl. (9) 23 (1944), 65 – 76 (French). · Zbl 0060.39508
[5] Ryszard Engelking, Topologia ogólna, Państwowe Wydawnictwo Naukowe, Warsaw, 1975 (Polish). Biblioteka Matematyczna. Tom 47. [Mathematics Library. Vol. 47]. Ryszard Engelking, General topology, PWN — Polish Scientific Publishers, Warsaw, 1977. Translated from the Polish by the author; Monografie Matematyczne, Tom 60. [Mathematical Monographs, Vol. 60].
[6] Leonard Gillman and Meyer Jerison, Rings of continuous functions, The University Series in Higher Mathematics, D. Van Nostrand Co., Inc., Princeton, N.J.-Toronto-London-New York, 1960. · Zbl 0093.30001
[7] Anthony W. Hager, Real-valued functions on Alexandroff (zero-set) spaces, Comment. Math. Univ. Carolinae 16 (1975), no. 4, 755 – 769. · Zbl 0312.54022
[8] H. Hahn, Über halbstetige und unstetige Funktionen, Sitzungsber. Akad. Wiss. Wien Abt. IIa 126 (1917), 91-110. · JFM 46.0399.04
[9] M. Katětov, On real-valued functions in topological spaces, Fund. Math. 38 (1951), 85 – 91. · Zbl 0045.25704
[10] M. Katětov, Correction to ”On real-valued functions in topological spaces” (Fund. Math. 38 (1951), pp. 85 – 91), Fund. Math. 40 (1953), 203 – 205. · Zbl 0045.25704
[11] Ernest P. Lane, Insertion of a continuous function, The Proceedings of the 1979 Topology Conference (Ohio Univ., Athens, Ohio, 1979), 1979, pp. 463 – 478 (1980). · Zbl 0443.54012
[12] Ernest P. Lane, Lebesgue sets and insertion of a continuous function, Proc. Amer. Math. Soc. 87 (1983), no. 3, 539 – 542. · Zbl 0511.54010
[13] R. Daniel Mauldin, On the Baire system generated by a linear lattice of functions, Fund. Math. 68 (1970), 51 – 59. · Zbl 0197.38104
[14] S. Mrówka, On some approximation theorems, Nieuw Arch. Wisk. (3) 16 (1968), 94 – 111.
[15] Hing Tong, Some characterizations of normal and perfectly normal spaces, Duke Math. J. 19 (1952), 289 – 292. · Zbl 0046.16203
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