Connected plane sets which contain no nondegenerate connected simple graph
Author:
B. D. Garrett
Journal:
Proc. Amer. Math. Soc. 113 (1991), 451459
MSC:
Primary 54D05; Secondary 54C60, 54C65
MathSciNet review:
991696
Fulltext PDF Free Access
Abstract 
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Abstract: Open questions are answered by showing that, if for each vertical line in the plane, is a cardinal number, there is a connected subset of the plane such that, if is a vertical line, then has cardinality and, if is a nondegenerate connected subset of , then contains two points of some vertical line.
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 A. M. Bruckner and J. Ceder, On jumping functions by connected sets, Czech. Math. J. 22 (1972), 435448. MR 0311843 (47:405)
 [2]
 J. Ceder, On Darboux selections, Fund. Math. 91 (1976), 8591. MR 0418032 (54:6076)
 [3]
 B. D. Garrett, D. Nelms, and K. R. Kellum, Characterizations of connected real functions, Jahr. Deutsch. Math. 73 (1971), 131137. MR 0486349 (58:6098)
 [4]
 F. B. Jones, Connected functions in the connected union of functions, presented at the Symposium on Pure and Applied Mathematics in Memory of Pasquale Porcelli, University of Houston, November, 1974.
 [5]
 , Connected simple graphs and a selection problem, Czech. Math. J. 25 (1975), 300301. MR 0369627 (51:5859)
 [6]
 M. E. Rudin, Lectures on set theoretic topology, Regional Conference Series in Mathematics, no. 23, Amer. Math. Soc., Providence, R.I., 1974. MR 0367886 (51:4128)
 [7]
 T. Tanaka, A question of Ceder concerning connected selections, Questions Answers Gen. Top. 1 (1983), 144148. MR 722097 (86h:54025)
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00029939199109916969
PII:
S 00029939(1991)09916969
Keywords:
Simple graph,
selection,
connectivity maps
Article copyright:
© Copyright 1991 American Mathematical Society
