On rings of Baire one functions

A. Deb Ray, Atanu Mondal

Abstract

This paper introduces the ring of all real valued Baire one functions, denoted by B1(X) and also the ring of all real valued bounded Baire one functions, denoted by B1(X). Though the resemblance between C(X) and B1(X) is the focal theme of this paper, it is observed that unlike C(X) and C(X) (real valued bounded continuous functions), B1 (X) is a proper subclass of B1(X) in almost every non-trivial situation. Introducing B1-embedding and B1-embedding, several analogous results, especially, an analogue of Urysohn’s extension theorem is established.


Keywords

B1(X); B∗1(X); zero set of a Baire one function; completely separated by B1(X); B1-embedded; B∗1-embedded

Subject classification

26A21; 54C30; 54C45; 54C50.

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References

J. P. Fenecios and E. A. Cabral, On some properties of Baire-1 functions, Int. Journal of Math. Analysis 7, no. 8 (2013), 393-402. https://doi.org/10.12988/ijma.2013.13035

I. Gelfand and A. Kolmogoroff, On rings of continuous functions on topological spaces, Dokl. Akad. Nauk SSSR 22 (1939), 11-15.

L. Gillman and M. Jerison, Rings of Continuous Functions, New York: Van Nostrand Reinhold Co., 1960. https://doi.org/10.1007/978-1-4615-7819-2

E. Hewitt, Rings of real-valued continuous functions, I, Trans. Amer. Math. Soc. 64 (1948), 54-99. https://doi.org/10.1090/S0002-9947-1948-0026239-9

P. Y. Lee, W. K. Tang and D. Zhao, An equivalent definition of functions of the first Baire class, Proc. Amer. Math. Soc. 129, no. 8 (2000), 2273-2275. https://doi.org/10.1090/S0002-9939-00-05826-3

M. H. Stone, Applications of the theory of Boolean rings to general topology, Trans. Amer. Math. Soc. 41 (1937), 375-481. https://doi.org/10.1090/S0002-9947-1937-1501905-7

L. Vesely, Characterization of Baire-one functions between topological spaces, Acta Universitatis Carolinae Mathematica et Physica 33, no. 2 (1992), 143-156.

D. Zhao, Functions whose composition with Baire class one functions are Baire class one, Soochow Journal of Mathematics 33, no. 4 (2007), 543-551.

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1. Ideals in B1(X) and residue class rings of B1(X) modulo an ideal
A. Deb Ray, Atanu Mondal
Applied General Topology  vol: 20  issue: 2  first page: 379  year: 2019  
doi: 10.4995/agt.2019.11417



Esta revista se publica bajo una licencia de Creative Commons Reconocimiento-NoComercial-SinObraDerivada 4.0 Internacional.

Universitat Politècnica de València

e-ISSN: 1989-4147   https://doi.org/10.4995/agt