A characterization of continua that contain no -ods and no -sets

Author:
Eldon J. Vought

Journal:
Proc. Amer. Math. Soc. **109** (1990), 545-551

MSC:
Primary 54F20; Secondary 54F25

MathSciNet review:
1012940

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Abstract: A proper, nondegenerate subcontinuum of a continuum is said to be a -set if, for every continuum and map of onto , some subcontinuum of is mapped by onto . Jim Davis asked whether a simple closed curve is the only atriodic continuum that contains no -set. An affirmative answer is given to this question. The result follows as a corollary to the more general theorem that a continuum contains no -od and has no -set if and only if it is a graph in which every point is contained in a simple closed curve. Properties of this class of graphs are also described.

**[1]**E. E. Grace and Eldon J. Vought,*Monotone decompositions of 𝜃_{𝑛}-continua*, Trans. Amer. Math. Soc.**263**(1981), no. 1, 261–270. MR**590423**, 10.1090/S0002-9947-1981-0590423-5**[2]**R. L. Moore,*Foundations of point set theory*, Revised edition. American Mathematical Society Colloquium Publications, Vol. XIII, American Mathematical Society, Providence, R.I., 1962. MR**0150722****[3]**Van C. Nall,*Weak confluence and 𝑊-sets*, Proceedings of the 1983 topology conference (Houston, Tex., 1983), 1983, pp. 161–193. MR**738474****[4]**Eldon J. Vought,*𝜔-connected continua and Jones’ 𝐾 function*, Proc. Amer. Math. Soc.**91**(1984), no. 4, 633–636. MR**746104**, 10.1090/S0002-9939-1984-0746104-6

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DOI:
https://doi.org/10.1090/S0002-9939-1990-1012940-7

Keywords:
Continuum,
-set,
-od,
atriodic,
class ,
-continuum,
graph,
monotone upper-semicontinuous decomposition,
quotient space

Article copyright:
© Copyright 1990
American Mathematical Society