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 , the cluster set of at is the set of all such that there exists a filter on converging to and the filter generated by converges to . The inverse cluster set of at is the set of all such that belongs to the cluster set of at . 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