The Mazurkiewicz distance and sets that are finitely connected at the boundary

Abstract

We study local connectedness, local accessibility and finite connectedness at the boundary, in relation to the compactness of the Mazurkiewicz completion of a bounded domain in a metric space. For countably connected planar domains we obtain a complete characterization. It is also shown exactly which parts of this characterization fail in higher dimensions and in metric spaces.

22 pages

References in corpus (2)

• Prime ends for domains in metric spaces
• The Dirichlet problem for p-harmonic functions with respect to the Mazurkiewicz boundary

Cited by in corpus (4)

• The Dirichlet problem for p-harmonic functions with respect to the Mazurkiewicz boundary
• Sphericalization and p-harmonic functions on unbounded domains in Ahlfors regular metric spaces
• Prime ends in metric spaces and boundary extensions of mappings
• Geometry of prime end boundary and the Dirichlet problem for bounded domains in metric spaces