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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 

 

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.


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DOI: https://doi.org/10.1090/S0002-9947-1972-0296308-0
Keywords: Cluster set, arc cluster set, arcwise connectedness
Article copyright: © Copyright 1972 American Mathematical Society