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

DOI:
https://doi.org/10.1090/S0002-9939-1991-0991696-9

MathSciNet review:
991696

Full-text PDF

Abstract | References | Similar Articles | Additional Information

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.

**[1]**A. M. Bruckner and J. Ceder,*On jumping functions by connected sets*, Czechoslovak Math. J.**22(97)**(1972), 435–448. MR**0311843****[2]**Jack Ceder,*On Darboux selections*, Fund. Math.**91**(1976), no. 2, 85–91. MR**0418032**, https://doi.org/10.4064/fm-91-2-85-91**[3]**B. D. Garrett, D. Nelms, and K. R. Kellum,*Characterizations of connected real functions*. part 1, Jber. Deutsch. Math.-Verein.**73**(1971/72), no. part 1, 131–137. MR**0486349****[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]**F. B. Jones,*Connected simple graphs and a selection problem*, Czechoslovak Math. J.**25(100)**(1975), 300–301. MR**0369627****[6]**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****[7]**Tadashi Tanaka,*A question of J. Ceder on connected selections*, Questions Answers Gen. Topology**1**(1983), no. 2, 144–148. MR**722097**

Retrieve articles in *Proceedings of the American Mathematical Society*
with MSC:
54D05,
54C60,
54C65

Retrieve articles in all journals with MSC: 54D05, 54C60, 54C65

Additional Information

DOI:
https://doi.org/10.1090/S0002-9939-1991-0991696-9

Keywords:
Simple graph,
selection,
connectivity maps

Article copyright:
© Copyright 1991
American Mathematical Society