The Open-Point and Compact-Open Topology on $C(X)$

Abstract:

      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.