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Connected plane sets which contain no nondegenerate connected simple graph

Author: B. D. Garrett
Journal: Proc. Amer. Math. Soc. 113 (1991), 451-459
MSC: Primary 54D05; Secondary 54C60, 54C65
MathSciNet review: 991696
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Abstract: Open questions are answered by showing that, if for each vertical line $L$ in the plane, $1 < {k_L} \leq c$ is a cardinal number, there is a connected subset $H$ of the plane such that, if $L$ is a vertical line, then $L \cap H$ has cardinality ${k_L}$ and, if $g$ is a nondegenerate connected subset of $H$, then $g$ contains two points of some vertical line.

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  • A. M. Bruckner and J. Ceder, On jumping functions by connected sets, Czechoslovak Math. J. 22(97) (1972), 435–448. MR 311843
  • Jack Ceder, On Darboux selections, Fund. Math. 91 (1976), no. 2, 85–91. MR 418032, DOI
  • B. D. Garrett, D. Nelms, and K. R. Kellum, Characterizations of connected real functions. 1, Jber. Deutsch. Math.-Verein. 73 (1971/72), no. part, 131–137. MR 486349
  • 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.
  • F. B. Jones, Connected simple graphs and a selection problem, Czechoslovak Math. J. 25(100) (1975), 300–301. MR 369627
  • Mary Ellen Rudin, Lectures on set theoretic topology, Published for the Conference Board of the Mathematical Sciences by the American Mathematical Society, Providence, R.I., 1975. Expository lectures from the CBMS Regional Conference held at the University of Wyoming, Laramie, Wyo., August 12–16, 1974; Conference Board of the Mathematical Sciences Regional Conference Series in Mathematics, No. 23. MR 0367886
  • Tadashi Tanaka, A question of J. Ceder on connected selections, Questions Answers Gen. Topology 1 (1983), no. 2, 144–148. MR 722097

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Keywords: Simple graph, selection, connectivity maps
Article copyright: © Copyright 1991 American Mathematical Society