# Some Properties of Thinness and Fine Topology with Relative Capacity

## TL;DR

This paper explores the concept of thinness in relation to a specific relative capacity within weighted variable exponent Sobolev spaces, examining its properties, connections with fine topology, and implications for potential theory.

## Contribution

It introduces a new notion of thinness based on relative capacity, analyzes its properties, and compares the fine topology with the Euclidean topology in the context of potential theory.

## Key findings

- Thinness relates to the structure of weighted variable exponent Sobolev spaces.
- Fine topology differs from Euclidean topology and is significant in potential theory.
- Properties of finely open and closed sets are characterized in this framework.

## Abstract

In this paper, we introduce a thinness in sense to a type of relative capacity for weighted variable exponent Sobolev space. Moreover, we reveal some properties of this thinness and consider the relationship with finely open and finely closed sets. We discuss fine topology and compare this topology with Euclidean one. Finally, we give some information about importance of the fine topology in the potential theory.