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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 

 

The compactness of the set of arc cluster sets of an arbitrary function


Author: John T. Gresser
Journal: Proc. Amer. Math. Soc. 37 (1973), 195-200
MSC: Primary 30A72
DOI: https://doi.org/10.1090/S0002-9939-1973-0318491-7
MathSciNet review: 0318491
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Abstract: It is known that if f is a continuous complex-valued function defined in the open unit disk D, then the set $ {\mathfrak{C}_f}(\zeta )\;(\zeta \in \partial D)$ of all arc cluster sets of f at $ \zeta $ is compact in a natural topology for all but at most a countable number of points $ \zeta \in \partial D$. We show that if f is an arbitrary complex-valued function defined on an arbitrary subset Z of the plane, then $ {\mathfrak{C}_f}(p)$ is compact for all but at most a countable number of points $ p \in Z \cup \partial Z$.


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DOI: https://doi.org/10.1090/S0002-9939-1973-0318491-7
Keywords: Arc cluster set, set of arc cluster sets of an arbitrary function, selector of arcs, missing arc cluster set
Article copyright: © Copyright 1973 American Mathematical Society