Let us say that a $g$-function $g(n,x)$ on a space $X$ satisfies the condition ($*$) provided: If $\{x_n\}\to p\in X$ and $x_n\in g(n,y_n)$ for every $n\in N$, then $y_n\to p$. We prove that a $k$-space $X$ is a metrizable space (a metrizable space with property $ACF$) if and only if there exists a strongly decreasing $g$-function $g(n,x)$ on $X$ such that $\{\overline{g(n,x)}:x\in X\}$ is $CF$ ($\{g(n,x):x\in X\}$ is $CF^*$) in $X$ for every $n\in N$ and the condition ($*$) is satisfied. Our results give a partial answer to a question posed by Z. Yun, X. Yang and Y. Ge and a positive answer to a conjecture posed by S. Lin, respectively.