Inverse cluster sets

Authors:
T. R. Hamlett and Paul Long

Journal:
Proc. Amer. Math. Soc. **53** (1975), 470-476

MSC:
Primary 54A20; Secondary 54C10

MathSciNet review:
0388312

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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.

**[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**0278254****[2]**James Dugundji,*Topology*, Allyn and Bacon, Inc., Boston, Mass., 1966. MR**0193606****[3]**R. V. Fuller,*Relations among continuous and various non-continuous functions*, Pacific J. Math.**25**(1968), 495–509. MR**0227952****[4]**T. R. Hamlett,*Cluster sets in general topology*, J. London Math. Soc. (2)**12**(1975/76), no. 2, 192–198. MR**0388311****[5]**Paul E. Long,*Connected mappings*, Duke Math. J.**35**(1968), 677–682. MR**0234428****[6]**Paul E. Long,*Concerning semiconnected maps*, Proc. Amer. Math. Soc.**21**(1969), 117–118. MR**0236890**, 10.1090/S0002-9939-1969-0236890-8**[7]**J. D. Weston,*Some theorems on cluster sets*, J. London Math. Soc.**33**(1958), 435–441. MR**0100680****[8]**Stephen Willard,*General topology*, Addison-Wesley Publishing Co., Reading, Mass.-London-Don Mills, Ont., 1970. MR**0264581**

Retrieve articles in *Proceedings of the American Mathematical Society*
with MSC:
54A20,
54C10

Retrieve articles in all journals with MSC: 54A20, 54C10

Additional Information

DOI:
https://doi.org/10.1090/S0002-9939-1975-0388312-7

Article copyright:
© Copyright 1975
American Mathematical Society