Prime ends and local connectivity

Bibliographic data

Bull. London Math. Soc. 40 (2008), no. 4, 817 - 826.

Arxiv version (updated since publication): arxiv:math.GN/0309022.

Published version: Abstract, PDF.

Update: Prior work of Torhorst and Sarason

In March 2012, Donald Sarason kindly pointed out that Theorem 1.1 in this paper was proved by Marie Torhorst in 1918 in her dissertation (Über die Randmenge einfach-zusammenhängender ebener Gebiete); the result was published in Math. Z. in 1921. It appears that this result had largely been forgotten. Perhaps even more interestingly, her thesis/paper appears to contain the first statement and proof of the fact that the Riemann map from the unit disc to a simply-connected domain extends continuously if and only if the boundary is locally connected.

It turns out that Don himself wrote a paper with the exact same title as mine in the 1960s, which reproves Torhorst's result from the work of Ursell and Young, which is the same argument by which Theorem 1.1 is established in my paper. However, his paper was not accepted for publication at the time, as he explains:
  • I submitted the paper to the Michigan Math. J., then edited by George Piranian, the person who taught me about prime ends and much more about complex analysis. (George is one of my mathematical heroes.) George discussed the paper with Collingwood, one of his collaborators. Their conclusion was that interest in prime ends at the time was at such a low ebb that the paper was likely to be largely ignored.
  • I did publish an abstract of the paper in the Notices of the A.M.S. (Vol 16 (1969), p. 701). At the time the Notices published abstracts of talks given at society meetings, plus what I think were called by-title abstracts, which any member of the society could use to announce a result. If my memory is correct, I received as a result of the abstract only one request for a copy of the paper.
Don's 1960s manuscript, along with George Piranian's letter and the announcement in the Notices, are contained in this PDF file, which he has kindly allowed me to make available.
To my knowledge, Theorem 1.3, which a characterization of local connectivity at a point and from which Theorem 1.1 follows using the Ursell-Young result, has not previously appeared elsewhere. (Note, however, that the argument that proves the "only if" direction is the same as the one that appears already in Don's paper, which also contains the "if" direction in the special case that every prime end whose impression contains the point in question is of the first kind.)

The arXiv version has been updated to explain this story, and contain more background on Marie Torhorst's work.