A theorem on fine connectedness (Q1977886)

An open connected set in the \(p\)-fine topology, i.e. in the coarsest topology making all \(p\)-superharmonic functions continuous, is called a \(p\)-fine domain. The purpose of this note is to prove the following theorem. Let \(\Omega\subset{\mathbb R}^n\) be a \(p\)-fine domain for \(1<p\leq n\) and let \(E\subset{\mathbb R}^n\) be a \(p\)-polar. Then \(\Omega\setminus E\) is a \(p\)-fine domain. As an application of the main result, the author establishes a general version of minimum principle.