The OpenPoint and CompactOpen 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). 

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Abstract: 
In this note we define a new topology on $C(X)$, the set of all realvalued 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. 
Citation: 
DOI:10.3770/j.issn:20952651.2020.03.007 
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