The open-point and compact-open topology on $C(X)$
Received:June 03, 2019  Revised:August 13, 2019
Key Word: $C_p(X)$, $C_k(X)$, $C_{kh}(X)$, $G_{\delta}$-dense
Fund ProjectL:Research supported by the National Natural Science Foundation of
 Author Name Affiliation E-mail Liang-Xue Peng College of Applied Science, Beijing University of Technology pengliangxue@bjut.edu.cn Yuan Sun Beijing University of Technology sy121690@126.com
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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: $[f, C, \varepsilon]=\{g\in C(X): |f(x)-g(x)|&lt;\varepsilon$ for every $x\in A\}$ 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 $X\}$, $\epsilon&gt;0$, while $U$ is an open subset of $X$ and $r\in \mathbb{R}$. The space $C(X)$ equipped with the new topology ${\mathcal T}_{kh}$ 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: \begin{enumerate} \item $X_0$ is $G_\delta$-dense in $X$; \item $C_{kh}(X)$ is regular; \item $C_{kh}(X)$ is Tychonoff; \item $C_{kh}(X)$ is a topological group. \end{enumerate} 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.