# zbMATH — the first resource for mathematics

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
Full Text:
##### References:
 [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
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.