Prime Ends and Local Connectivity

Abstract

Let U be a simply connected domain on the Riemann sphere whose complement K contains more than one point. We establish a characterization of local connectivity of K at a point in terms of the prime ends whose impressions contain this point. Invoking a result of Ursell and Young, we obtain an alternative proof of a theorem of Torhorst, which states that the impression of a prime end of $U$ contains at most two points at which $K$ is locally connected.

13 pages, 2 figures. V6: Added additional remarks on the mathematical work of Marie Torhorst. Summary of previous versions: V5 - Updated the article by adding a historical note. V4 - Final preprint prior to publication. V3 - Proof of the Ursell-Young theorem and illustrations added. V2 - The original short note was expanded to a full article. V1 - original short note

References in corpus (1)

• Prime Ends and Local Connectivity

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