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Proceedings of the American Mathematical Society

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

 

 

Inverse cluster sets


Authors: T. R. Hamlett and Paul Long
Journal: Proc. Amer. Math. Soc. 53 (1975), 470-476
MSC: Primary 54A20; Secondary 54C10
DOI: https://doi.org/10.1090/S0002-9939-1975-0388312-7
MathSciNet review: 0388312
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Abstract: For a function $ f:X \to Y$, the cluster set of $ f$ at $ x\epsilon X$ is the set of all $ y\epsilon Y$ such that there exists a filter $ \mathcal{F}$ on $ X$ converging to $ x$ and the filter generated by $ f(\mathcal{F})$ converges to $ y$. The inverse cluster set of $ f$ at $ y\epsilon Y$ is the set of all $ x\epsilon X$ such that $ y$ belongs to the cluster set of $ f$ at $ x$. General properties of inverse cluster sets are proved, including a necessary and sufficient condition for continuity. Necessary and sufficient conditions for functions to have a closed graph in terms of inverse cluster sets are also given. Finally, a known theorem giving a condition as to when a connected function is also a connectivity function is generalized and further investigated in terms of inverse cluster sets.


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DOI: https://doi.org/10.1090/S0002-9939-1975-0388312-7
Article copyright: © Copyright 1975 American Mathematical Society