关于环$B_1(X)$的结构空间
On Structure space of the ring $B_1(X)$
摘要：
本文继续研究拓扑空间$(X,τ)$上的Baire 1函数环，记为$B_1(X)$，推广了著名的M. H.从$C(X)$到$B_1(X)$的Stones定理。引入$B_1(X)$的结构空间，建立了Gelfand Kolmogoroff定理的一个类似.观察到$(X,τ)$可能未嵌入$B_1(X)$的结构空间内。这一观察启发我们引入了一种较弱的嵌入形式，并表明在$X$是$T_4$空间的情况下，$X$作为稠密子空间弱嵌入在$B_1(X)$的结构空间中。进一步证明了由所有有界Baire 1函数构成的环$B_1^{*}(X)$是C-型环，并且$B_1^{*}(X)$的结构空间与$B_1(X)$的结构空间同胚.在$X$上引入一个比原$T_4$拓扑$τ$更精细的拓扑$σ$，证明了如果$σ$严格精细于$τ$，则$B_1(X)$包含自由（极大）理想.证明了$τ= σ$当且仅当$B_1(X) = C(X)$ .此外，在所有完美正规$T_1$空间类中，$B_1(X) = C(X)$等价于空间$X$的离散性。
英文摘要：
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 $X$.
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