A new Cartan-type property and strict quasicoverings when $p=1$ in metric spaces

TL;DR

This paper establishes a new Cartan-type property for the fine topology at p=1 in certain metric spaces, enabling the construction of strict quasicoverings and advancing the understanding of fine Newton-Sobolev spaces.

Contribution

It introduces a novel Cartan-type property for the fine topology at p=1 and applies it to construct strict quasicoverings and analyze fine Newton-Sobolev spaces.

Findings

Proved a new Cartan-type property for the fine topology at p=1.

Constructed strict quasicoverings of 1-finely open sets.

Analyzed fine Newton-Sobolev spaces on 1-finely open sets.

Abstract

In a complete metric space that is equipped with a doubling measure and supports a Poincar\'e inequality, we prove a new Cartan-type property for the fine topology in the case $p=1$. Then we use this property to prove the existence of $1$-finely open \emph{strict subsets} and \emph{strict quasicoverings} of $1$-finely open sets. As an application, we study fine Newton-Sobolev spaces in the case $p=1$, that is, Newton-Sobolev spaces defined on $1$-finely open sets.