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

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

MathSciNet review:
1012940

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Abstract | References | Similar Articles | Additional Information

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*-continua, Trans. Amer. Math. Soc.**263**(1981), 261-270. MR**590423 (81k:54058)****[2]**R. L. Moore,*Foundations of point set theory*, Amer. Math. Soc. Colloq. Publ.**13**(1962). MR**0150722 (27:709)****[3]**V. C. Nail,*Weak confluence and**-sets*, Topology Proc.**8**(1983), 161-193. MR**738474 (85d:54042)****[4]**Eldon J. Vought,*-connected continua and Jones'**function*, Proc. Amer. Math. Soc.**91**(1984), 633-636. MR**746104 (86d:54055)**

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Additional Information

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