Connectedness in the Pluri-fine Topology

Abstract

We study connectedness in the pluri-fine topology on $\CC^n$ and obtain the following results. If $\Omega$ is a pluri-finely open and pluri-finely connected set in $\CC^n$ and $E\subset\CC^n$ is pluripolar, then $\Omega\setminus E$ is pluri-finely connected. The proof hinges on precise information about the structure of open sets in the pluri-fine topology: Let $\Omega$ be a pluri-finely open subset of $\CC^{n}$. If $z$ is any point in $\Omega$, and $L$ is a complex line passing through $z$, then obviously $\Omega \cap L$ is a finely open neighborhood of $z$ in $L$. Now let $C_L$ denote the finely connected component of $z$ in $\Omega\cap L$. Then $\cup_{L\ni z} C_L$ is a pluri-finely connected neighborhood of $z$. As a consequence we find that if $v$ is a finely plurisubharmonic function defined on a pluri-finely connected pluri-finely open set, then $v= -\infty$ on a pluri-finely open subset implies $v\equiv -\infty$.