论文标题
在环的结构空间上$ b_1（x）$
On Structure space of the ring $B_1(X)$
论文作者
论文摘要
在本文中，我们继续研究Baire Ring One在拓扑空间$（x，τ）$上的功能，该功能由$ b_1（x）$表示，并将著名的M. H. Stones的定理从$ c（x）$扩展到$ b_1（x）$。建立了Gelfand Kolmogoroff定理的类似物$ B_1（X）$的结构空间。据观察，$（x，τ）$可能不会嵌入$ b_1（x）$的结构空间内。这种观察激发了我们引入较弱的嵌入形式，并表明，如果$ x $是$ t_4 $ space，则$ x $薄弱地嵌入为密集的子空间，在$ b_1（x）$的结构空间中。进一步确定的是，所有有限的baire一个函数的环$ b_1^{*}（x）$是C型环，而且$ b_1^{*}（x）$的结构空间与$ b_1（x）$的结构空间同源。引入更好的拓扑$σ$，比原始$ t_4 $ topology $τ$上的$ x $上的$ x $，如果$ b_1（x）$严格比$τ$，$ b_1（x）$包含免费（最大）理想。还证明，$τ=σ$，仅当$ b_1（x）= c（x）$。此外，在所有完美普通$ T_1 $空间的类中，$ b_1（x）= c（x）$等效于空间$ x $的离散性。
In this article, we continue our study of the ring of Baire one functions on a topological space $(X,τ)$, denoted by $B_1(X)$ and extend the well known M. H. Stones's theorem from $C(X)$ to $B_1(X)$. Introducing the structure space of $B_1(X)$, an analogue of Gelfand Kolmogoroff theorem is established. It is observed that $(X,τ)$ may not be embedded inside the structure space of $B_1(X)$. This observation inspired us to introduce a weaker form of embedding and show that in case $X$ is a $T_4$ space, $X$ is weakly embedded as a dense subspace, in the structure space of $B_1(X)$. It is further established that the ring $B_1^{*}(X)$ of all bounded Baire one functions is a C-type ring and also, the structure space of $B_1^{*}(X)$ is homeomorphic to the structure space of $B_1(X)$. Introducing a finer topology $σ$ than the original $T_4$ topology $τ$ on $X$, it is proved that $B_1(X)$ contains free (maximal) ideals if $σ$ is strictly finer than $τ$. It is also proved that $τ= σ$ if and only if $B_1(X) = C(X)$. Moreover, in the class of all perfectly normal $T_1$ spaces, $B_1(X) = C(X)$ is equivalent to the discreteness of the space $X$.
