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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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  • [1] M. P. Berri, J. R. Porter and R. M. Stephenson, Jr., A survey of minimal topological spaces, General Topology and its Relations to Modern Analysis and Algebra, III (Proc. Conf., Kanpur, 1968), Academia, Prague, 1971, pp. 93-114. MR 43 #3985. MR 0278254 (43:3985)
  • [2] J. Dugundji, Topology, Allyn and Bacon, Boston, Mass., 1966. MR 0193606 (33:1824)
  • [3] R. V. Fuller, Relations among continuous and various non-continuous functions, Pacific J. Math. 25 (1968), 495-509. MR 37 #3536. MR 0227952 (37:3536)
  • [4] T. R. Hamlett, Cluster sets in general topology, J. London Math. Soc. (to appear). MR 0388311 (52:9148)
  • [5] Paul E. Long, Connected mappings, Duke Math. J. 35 (1968), 677-682. MR 38 #2745. MR 0234428 (38:2745)
  • [6] -, Concerning semiconnected maps, Proc. Amer. Math. Soc. 21 (1969), 117-118. MR 38 #5183. MR 0236890 (38:5183)
  • [7] J. D. Weston, Some theorems on cluster sets, J. London Math. Soc. 33 (1958), 435-441. MR 20 #7109. MR 0100680 (20:7109)
  • [8] Stephen Willard, General topology, Addison-Wesley, Reading, Mass., 1970. MR 41 #9173. MR 0264581 (41:9173)

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Additional Information

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

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