$C(X)$上的开点与紧开拓扑
The Open-Point and Compact-Open Topology on $C(X)$

DOI：10.3770/j.issn:2095-2651.2020.03.007

 作者 单位 彭良雪 北京工业大学应用数理学院, 北京 100124 孙愿 北京工业大学应用数理学院, 北京 100124

对于Tychonoff空间$X$,令$C(X)$为$X$上的所有连续实值函数构成的集合.在本文中我们在$C(X)$上定义了一个新的拓扑,这个拓扑的子基是由如下两种形式的集合构成的: $[f,C,\varepsilon]=\{g\in C(X): |f(x)-g(x)|<\varepsilon$，$x\in C\}$和$[U, r]^-=\{g\in C(X): g^{-1}(r)\cap U\neq\emptyset\}$,其中$f\in C(X)$, 集合$C$是$X$上的非空紧集,$\epsilon>0$, 同时$U$是$X$上的开子集, $r$是实数. 我们把具有此拓扑的实值连续函数空间$C(X)$记为$C_{kh}(X)$. 令$X_0=\{x\in X: x$是$X$中的孤立点\},$X_{c}=\{x\in X: x$在$X$中有紧邻域$\}$. 我们得到如下结论: 如果$X$是一个具有性质$X_0=X_c$的Tychonoff空间, 则下列性质等价: (1)~~$X_0$是$X$中的$G_\delta$-稠密集; (2)~~$C_{kh}(X)$是正则空间; (3)~~$C_{kh}(X)$是Tychonoff空间; (4)~~$C_{kh}(X)$是拓扑群. 我们还证明了如果$X$是一个具有性质$X_0=X_c$的Tychonoff空间且$C_{kh}(X)$是一个具有可数伪特征的正则空间, 则$X$是$\sigma$-紧空间. 另外, 如果$X$是一个可度量的半紧的可数空间, 则空间$C_{kh}(X)$是第一可数空间.

In this note we define a new topology on $C(X)$, the set of all real-valued continuous functions on a Tychonoff space $X$. The new topology on $C(X)$ is the topology having subbase open sets of both kinds: $[f, C, \varepsilon]=\{g\in C(X): |f(x)-g(x)|<\varepsilon$ for every $x\in C\}$ and $[U, r]^-=\{g\in C(X): g^{-1}(r)\cap U\neq\emptyset\}$, where $f\in C(X)$, $C\in {\mathcal K}(X)=\{$ nonempty compact subsets of $X\}$, $\epsilon>0$, while $U$ is an open subset of $X$ and $r\in \mathbb{R}$. The space $C(X)$ equipped with the new topology ${\mathcal T}_{kh}$ which is stated above is denoted by $C_{kh}(X)$. Denote $X_0=\{x\in X: x$ is an isolated point of $X$\} and $X_{c}=\{x\in X: x$ has a compact neighborhood in $X\}$. We show that if $X$ is a Tychonoff space such that $X_0=X_c$, then the following statements are equivalent: (1)\ \ $X_0$ is $G_\delta$-dense in $X$; (2)\ \ $C_{kh}(X)$ is regular; (3)\ \ $C_{kh}(X)$ is Tychonoff; (4)\ \ $C_{kh}(X)$ is a topological group. We also show that if $X$ is a Tychonoff space such that $X_0=X_c$ and $C_{kh}(X)$ is regular space with countable pseudocharacter, then $X$ is $\sigma$-compact. If $X$ is a metrizable hemicompact countable space, then $C_{kh}(X)$ is first countable.