论文标题
有关戒指的更多信息$ b_1（x）$和$ b_1^*（x）$
More on the rings $B_1(X)$ and $B_1^*(X)$
论文作者
论文摘要
本文主要集中在拓扑空间上所有有界的Baire的环上。统一规范拓扑是由所有有界Baire一个功能的集合中定义的$ \ sup $ -norm引起的。关于此拓扑，$ b_1^*（x）$是拓扑环。事实证明，在统一的规范拓扑下，所有单元的集合形成了一个开放式集合，因此，$ b_1^*（x）$的每个最大理想都在$ b_1^*（x）$中，具有统一的规范拓扑。由于$ b_1（x）$上的统一规范拓扑的自然扩展是$ b_1^*（x）\ neq b_1（x）$的自然扩展，但没有显示这些功能，因此在$ b_1（x）上定义了一种称为$ m_b $ - 学位的拓扑，以在$ b_1（x）上实现这些结果，以在$ b_1（x）$上实现这些结果。事实证明，相对$ m_b $拓扑与$ b_1^*（x）$的统一规范拓扑相吻合，并且仅当$ b_1（x）= b_1^*（x）$。此外，当$ b_1（x）$带有$ m_b $ -topology时，当时仅当$ b_1（x）= b_1^*（x）$时，仅是第1次计数。 \\本文的最后一部分建立了$ b_1^*（x）$的理想与$ z_b $ -filters的特殊类别之间的对应关系，称为$ e_b $ -filters在普通拓扑空间$ x $上。还可以观察到，对于普通空间，所有最大理想的基础性$ b_1（x）$，而$ b_1^*（x）$的基础性是相同的。
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.
