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.
Funding Agencies|Swedish Research Council; Swedish Fulbright Commission; Charles Phelps Taft Research Center at the University of Cincinnati; Taft Research Center of the University of Cincinnati; Simons Foundation [200474]; NSF [DMS-1200915]