More on the rings $B_1(X)$ and $B_1^{\ast }(X)$

Abstract

This paper focuses mainly on the ring of all bounded Baire one functions on a topological space. The uniform norm topology arises from the $\sup$-norm defined on the collection $B_1^{\ast }(X)$ of all bounded Baire one functions. With respect to this topology, $B_1^{\ast }(X)$ is a topological ring. It is proved that under uniform norm topology, the set of all units forms an open set and as a consequence of it, every maximal ideal of $B_1^{\ast }(X)$ is closed in $B_1^{\ast }(X)$ with uniform norm topology. Since the natural extension of uniform norm topology on $B_1(X)$, when $B_1^{\ast }(X)\ne B_1(X)$, does not show up these features, a topology called $m_B$-topology is defined on $B_1(X)$ suitably to achieve these results on $B_1(X)$. It is proved that the relative $m_B$ topology coincides with the uniform norm topology on $B_1^{\ast }(X)$ if and only if $B_1(X)=B_1^{\ast }(X)$. Moreover, $B_1(X)$ with $m_B$-topology is 1st countable if and only if $B_1(X)=B_1^{\ast }(X)$. \\ The last part of the paper establishes a correspondence between the ideals of $B_1^{\ast }(X)$ and a special class of $Z_B$-filters, called $e_B$-filters on a normal topological space $X$. It is also observed that for normal spaces, the cardinality of the collection of all maximal ideals of $B_1(X)$ and those of $B_1^{\ast }(X)$ are the same.