论文标题
在与H-seber空间有关的功能空间上
On Function Spaces Related to H-sober Spaces
论文作者
论文摘要
在本文中，我们主要研究与H-苏伯空间相关的功能空间。对于不可修复的子集系统h和$ t_ {0} $ spaces $ x $和$ y $，证明$ y $是h-seber如果功能空间$ \ mathbb {c}（x，x，y）$的所有连续函数$ f：x \ longrightArrow y $配备了h longrightArrow y $，配备了pointwissiswise contrace factor factor factig factirben $ \ Mathbb {C}（X，Y）$配备了ISBELL拓扑。一种立即的推论是，对于$ t_ {0} $ x $，$ y $是一个清醒的空间（分别，$ d $ - 空间，滤过良好的空间），如果功能空间$ \ mathbb {c}（c}（x，y）$，配备了pointwise convergence的拓平空间（$ d $ d $ d $ -dspace） $ \ Mathbb {c}（x，y）$配备了ISBELL拓扑的空间（分别，$ d $ - 空间，过滤良好的空间）。结果表明，如果功能空间$ \ mathbb {c}（x，y）$配备了紧凑型拓扑，则$ t_ {0} $ spaces $ x $和$ y $，则$ y $是h-sober。还讨论了配备了Scott拓扑的功能空间$ \ MATHBB {C}（X，Y）$。
In this paper, we mainly study the function spaces related to H-sober spaces. For an irreducible subset system H and $T_{0}$ spaces $X$ and $Y$, it is proved that $Y$ is H-sober iff the function space $\mathbb{C}(X, Y)$ of all continuous functions $f : X\longrightarrow Y$ equipped with the topology of pointwise convergence is H-sober iff the function space $\mathbb{C}(X, Y)$ equipped with the Isbell topology is H-sober. One immediate corollary is that for a $T_{0}$ space $X$, $Y$ is a sober space (resp., $d$-space, well-filtered space) iff the function space $\mathbb{C}(X, Y)$ equipped with the topology of pointwise convergence is a sober space (resp., $d$-space, well-filtered space) iff the function space $\mathbb{C}(X, Y)$ equipped with the the Isbell topology is a sober space (resp., $d$-space, well-filtered space). It is shown that $T_{0}$ spaces $X$ and $Y$, if the function space $\mathbb{C}(X, Y)$ equipped with the compact-open topology is H-sober, then $Y$ is H-sober. The function space $\mathbb{C}(X, Y)$ equipped with the Scott topology is also discussed.
