On function spaces for general topological spaces

In this paper we mean by a space a topological space with no separation axiom unless otherwise specified, and we denote by $R$ and $I$ the real line and the closed unit interval respectively.Given two spaces $X$ and $Y$ , let $F(X$, }' $)$ denote the set of all maps from $X$ into $Y,$ $C(X, Y)$ the set of all continuous maps from $X$ into $Y$ .In case $Y$ is the real line $R,$ $C(X, R)$ is denoted more simply by $C(X)$ .The map $\rho:is bijective; this correspondence is called the exponential map.A topology on $C(X, T)$ is called proper if for every space $Y$ and any $f\in C(X\times Y, T)$ the map $\rho(f)$ belongs to $C(Y, C(X, T))$ .Similarly, a topology on $C(X, T)$ is called admissible if for every space $Y$ and any $g\in C(Y, C(X, T))$ the map $\rho^{-1}(g)$ belongs to $C(X\times Y, T)$ .A topology on $C(X, T)$ that is both proper and admissible is called an acceptable topology (see [1], [2] and [3]).As is well known, the compact-open topology on $C(X, T)$ is acceptable for any space $T$ when $X$ is locally compact Hausdorff (see [4]).Furthermore, the following theorem was proved by R. Arens [1].THEOREM 1.1.Let $X$ be a Tychonoff space.Then the following conditions are equivalent.(1) $X$ is locally compact.(2) There exists an acceptable topology on $C(X)$ .In the case that $X$ is not necessarily Tychonoff, Professor T. Ishii raised the following problem: Characterize a space $X$ such that there exists an acceptable topology on $C(X)$ .The main purpose of this paper is to give the solution for this problem by proving the following theorem.