Quasiopen and p-path open sets, and characterizations of quasicontinuity

TL;DR

This paper characterizes quasiopen sets and quasicontinuous functions in metric spaces, showing their equivalence under certain conditions and establishing the quasicontinuity of Newton-Sobolev functions on quasiopen sets.

Contribution

It provides new characterizations of quasiopen sets and quasicontinuity, and proves the equivalence of quasiopen and p-path open sets in specific metric spaces.

Findings

Quasiopen and p-path open sets coincide in certain metric spaces.

Newton-Sobolev functions on quasiopen sets are quasicontinuous.

Characterizations of quasiopen sets and quasicontinuity are established.

Abstract

In this paper we give various characterizations of quasiopen sets and quasicontinuous functions on metric spaces. For complete metric spaces equipped with a doubling measure supporting a p-Poincar\'e inequality we show that quasiopen and p-path open sets coincide. Under the same assumptions we show that all Newton-Sobolev functions on quasiopen sets are quasicontinuous.