The Open-Point and Compact-Open Topology on $C(X)$
Received:June 03, 2019  Revised:October 09, 2019
Key Word: $C_p(X)$   $C_k(X)$   $C_{kh}(X)$   $G_{\delta}$-dense
Fund ProjectL:Supported by the National Natural Science Foundation of China (Grant No.11771029) and the Natural Science Foundation of Beijing City (Grant No.1202003).
 Author Name Affiliation Liangxue PENG College of Applied Science, Beijing University of Technology, Beijing 100124, P. R. China Yuan SUN College of Applied Science, Beijing University of Technology, Beijing 100124, P. R. China
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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: $[f, C, \varepsilon]=\{g\in C(X): |f(x)-g(x)|<\varepsilon$ 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.