More on the rings \(B_1(X)\) and \(B_1^(X)\)

• Published in: arXiv.org 2023-06
• Main Authors: Mondal, Atanu, A Deb Ray
• Format: Article
• Language: English
• Subjects: Collection; Rings (mathematics); Topology
• ISSN: 2331-8422

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 \(B_1^*(X)\) of all bounded Baire one functions. With respect to this topology, \(B_1^*(X)\) 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 \(B_1^*(X)\) is closed in \(B_1^*(X)\) with uniform norm topology. Since the natural extension of uniform norm topology on \(B_1(X)\), when \(B_1^*(X) \neq B_1(X)\), 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 \(B_1^*(X)\) if and only if \(B_1(X) = B_1^*(X)\). Moreover, \(B_1(X)\) with \(m_B\)-topology is 1st countable if and only if \(B_1(X) = B_1^*(X)\). \\ The last part of the paper establishes a correspondence between the ideals of \(B_1^*(X)\) 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 \(B_1^*(X)\) are the same.