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

TL;DR

This paper characterizes when $ ext{psi}$-uniform and uniform domains in Banach spaces retain their properties after removing a countable, well-separated subset, establishing equivalences under specific separation conditions.

Contribution

It proves that $ ext{psi}$-uniform and uniform domains in Banach spaces preserve their properties after removing certain countable, well-separated subsets, under quasihyperbolic separation conditions.

Findings

$ ext{psi}$-uniform domains are preserved after removing $P$.

Uniform domains remain uniform after removing $P$.

Separation condition ensures property preservation.

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 $\frac{1}{2}$.