论文标题
$ x $的特征是区分$ c_p（x）$的$ x $，其应用程序
A characterization of $X$ for which spaces $C_p(X)$ are distinguished and its applications
论文作者
论文摘要
我们证明，在Tychonoff Space上的连续实现功能的本地凸空间$ C_ {p}（x）$在\ cite {knight}的意义上仅$ x $是$Δ$ - 空间时，才能区分配备了点上融合的拓扑的$ x $。作为此特征定理的应用，我们获得以下结果： 1）如果$ x $是一个（尤其是紧凑的）空间，以便区分$ c_p（x）$，则$ x $分散了。 2）对于ISBELL-MRówka类型$ X $的每个可分离紧凑空间，空间$ C_P（x）$都分为区分。 3）如果$ x $是列出$ [0，ω_1] $的紧凑空间，则没有区分$ c_p（x）$。 我们观察到，存在无法分离的Metrizable Space $ x $的存在，因此$ C_P（x）$是区分的，独立于ZFC。我们还探讨了一个问题，在基本拓扑操作下，$δ$空间的类别是不变的。
We prove that the locally convex space $C_{p}(X)$ of continuous real-valued functions on a Tychonoff space $X$ equipped with the topology of pointwise convergence is distinguished if and only if $X$ is a $Δ$-space in the sense of \cite {Knight}. As an application of this characterization theorem we obtain the following results: 1) If $X$ is a Čech-complete (in particular, compact) space such that $C_p(X)$ is distinguished, then $X$ is scattered. 2) For every separable compact space of the Isbell--Mrówka type $X$, the space $C_p(X)$ is distinguished. 3) If $X$ is the compact space of ordinals $[0,ω_1]$, then $C_p(X)$ is not distinguished. We observe that the existence of an uncountable separable metrizable space $X$ such that $C_p(X)$ is distinguished, is independent of ZFC. We explore also the question to which extent the class of $Δ$-spaces is invariant under basic topological operations.
