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Transactions of the American Mathematical Society

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The connectedness of the collection of arc cluster sets

Author: Peter Lappan
Journal: Trans. Amer. Math. Soc. 168 (1972), 303-310
MSC: Primary 30A72
MathSciNet review: 0296308
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Abstract: Let $ f$ be a continuous complex-valued function defined on the unit disk and let $ p$ be a boundary point of the disk. A very natural topology on the collection of all arc cluster sets of $ f$ at the point $ p$ has been investigated by Belna and Lappan [1] who proved that this collection is a compact set under certain suitable conditions. It is proved here that this collection is an arcwise connected set under the topology in question, but is not in general locally arcwise connected or even locally connected. It is also shown by example that it is generally not possible to map the real line onto the collection of arc cluster sets at $ p$ in a continuous manner.

References [Enhancements On Off] (What's this?)

  • [1] C. L. Belna and P. Lappan, The compactness of the set of arc cluster sets, Michigan Math. J. 16 (1969), 211-214. MR 9, 7111. MR 0245805 (39:7111)
  • [2] C. Kuratowski, Topologie. I. Espaces métrisables, espaces complets, 2ième éd., Monografie Mat., Tom 20, PWN, Warsaw, 1948. MR 10, 389. MR 0028007 (10:389b)
  • [3] P. Lappan, Continua which are curvilinear cluster sets, Nagoya Math. J. 34 (1969), 25-34. MR 39 #1660. MR 0240311 (39:1660)

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Keywords: Cluster set, arc cluster set, arcwise connectedness
Article copyright: © Copyright 1972 American Mathematical Society

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