Let $R$ be a ring with identity and let $g(x)$ be a polynomial in $Z(R)[x]$ where $Z(R)$ denotes the center of $R$. An element $r\in R$ is called $g(x)$-clean if $r=u+s$ for some $u,s\in R$ such that $u$ is a unit and $g(s)=0$. The ring $R$ is $g(x)$-clean if every element of $R$ is $g(x)$-clean. We consider $g(x)=x(x-c)$ where $c$ is a unit in $R$ such that every root of $g(x)$ is central in $R$. We show, via set-theoretic topology, that among conditions equivalent to $R$ being $g(x)$-clean, is that $R$ is right (left) $c$-topologically boolean.