The compactness of the set of arc cluster sets of an arbitrary function
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- by John T. Gresser
- Proc. Amer. Math. Soc. 37 (1973), 195-200
- DOI: https://doi.org/10.1090/S0002-9939-1973-0318491-7
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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 $p \in Z \cup \partial Z$.References
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- Charles Belna and Peter Lappan, The compactness of the set of arc cluster sets, Michigan Math. J. 16 (1969), 211–214. MR 245805
- E. F. Collingwood and A. J. Lohwater, The theory of cluster sets, Cambridge Tracts in Mathematics and Mathematical Physics, No. 56, Cambridge University Press, Cambridge, 1966. MR 231999
- Kiyoshi Noshiro, Cluster sets, Ergebnisse der Mathematik und ihrer Grenzgebiete, (N.F.), Heft 28, Springer-Verlag, Berlin-Göttingen-Heidelberg, 1960. MR 133464
Bibliographic Information
- © Copyright 1973 American Mathematical Society
- 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