On removability properties of $\psi$-uniform domains in Banach spaces

Abstract

Suppose that $E$ denotes a real Banach space with the dimension at least 2. The main aim of this paper is to show that a domain $D$ in $E$ is a $\psi$-uniform domain if and only if $D\backslash P$ is a $\psi_1$-uniform domain, and $D$ is a uniform domain if and only if $D\backslash P$ also is a uniform domain, whenever $P$ is a closed countable subset of $D$ satisfying a quasihyperbolic separation condition. This condition requires that the quasihyperbolic distance (w.r.t. $D$) between each pair of distinct points in $P$ has a lower bound greater than or equal to $\frac{1}{2}$.