The minimal support for a continuous functional on a function space Ⅱ

Abstract

Let $C_{k}(X)$ be the function space with the compact-open topology over a Tychonoff space $X$ and $\xi$ a continuous real-valued function on $C_{k}(X)$. A closed subset $S$ of $X$ is called a support for $\xi$ if $\xi(f)=\xi(g)$ holds for any pair $(f, g)$ of functions in $C_{k}(X)$ such that $f|_{s}=g|_{s}$. It is proved that the minimal support for any realvalued continuous function on the space $C_{k}(X)$ exists.