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

Abstract

In a complete metric space that is equipped with a doubling measure and supports a Poincaré 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 strict subsets and 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.