On Rings of Baire one functions

Abstract

This paper introduces the ring of all real valued Baire one functions, denoted by $B_1(X)$ and also the ring of all real valued bounded Baire one functions, denoted by $B_1^{\ast }(X)$. Though the resemblance between $C(X)$ and $B_1(X)$ is the focal theme of this paper, it is observed that unlike $C(X)$ and $C^{\ast }(X)$ (real valued bounded continuous functions), $B_1^{\ast }(X)$ is a proper subclass of $B_1(X)$ in almost every non-trivial situation. Introducing $B_1$-embedding and $B_1^{\ast }$-embedding, several analogous results, especially, an analogue of Urysohn's extension theorem is established.

11 pages