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The Mazurkiewicz Distance and Sets that are Finitely Connected at the Boundary

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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.

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Acknowledgments

This research was begun while the first two authors visited the University of Cincinnati during the first half year of 2010, and continued while the third author visited Linköpings universitet in March 2011, and during the stay of the three authors at Institut Mittag-Leffler in Autumn 2013. We wish to thank these institutions for their kind hospitality. We also wish to thank Tomasz Adamowicz and Harold Bell for fruitful discussions. The first two authors were supported by the Swedish Research Council. The first author was also a Fulbright scholar during his visit to the University of Cincinnati, supported by the Swedish Fulbright Commission, while the second author was a Visiting Taft Fellow during her visit to the University of Cincinnati, supported by the Charles Phelps Taft Research Center at the University of Cincinnati. The third author was also supported by the Taft Research Center of the University of Cincinnati and by grant #200474 from the Simons Foundation and NSF grant DMS-1200915.

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Authors and Affiliations

  1. Department of Mathematics, Linköpings universitet, 581 83, Linköping, Sweden

    Anders Björn & Jana Björn

  2. Department of Mathematical Sciences, University of Cincinnati, P.O. Box 210025, Cincinnati, OH, 45221-0025, USA

    Nageswari Shanmugalingam

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  1. Anders Björn
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  2. Jana Björn
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  3. Nageswari Shanmugalingam
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Correspondence to Nageswari Shanmugalingam.

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Björn, A., Björn, J. & Shanmugalingam, N. The Mazurkiewicz Distance and Sets that are Finitely Connected at the Boundary. J Geom Anal 26, 873–897 (2016). https://doi.org/10.1007/s12220-015-9575-9

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