On Burgess's theorem and related problems

Authors:
Hisao Kato and Xiangdong Ye

Journal:
Proc. Amer. Math. Soc. **128** (2000), 2501-2506

MSC (1991):
Primary 54B15, 54F15, 54F50

DOI:
https://doi.org/10.1090/S0002-9939-00-05247-3

Published electronically:
February 25, 2000

MathSciNet review:
1653402

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Abstract | References | Similar Articles | Additional Information

Abstract: Let be a graph. We determine all graphs which are -like. We also prove that if are graphs, then in order that each -like continuum be -indecomposable for some it is necessary and sufficient that if is a graph, then is not -like for some integer with . This generalizes a well known theorem of Burgess.

**[B1]**C.E. Burgess, Continua which are the sum of a finite number of indecomposable continua, Proc. Amer. Math. Soc.,**4**(1953), 234-239. MR**14:894a****[B2]**C.E. Burgess, Chainable continua and indecomposability, Pacific J. Math.,**9**(1959), 653-659. MR**22:1867****[K]**H. Kato, A note on refinable maps and quasi-homeomorphic compacta, Proc. Japan Acad.,**58**(1982), 69-71. MR**83d:54019****[LXY]**J. Lu, J. Xiong and X. Ye, The inverse limit space and the dynamics of a graph map, Preprint, 1997.**[MS]**S. Mardei and J. Segal, -mappings onto polyhedral, Trans. Amer. Math. Soc.,**109**(1963), 146-164. MR**28:1592****[N]**S. B. Nadler Jr., Continuum Theory, Pure and Appl. Math.,**158**(1992). MR**93m:54002**

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Additional Information

**Hisao Kato**

Affiliation:
Institute of Mathematics, University of Tsukuba, Tsukuba-Shi Ibaraki, 305, Japan

Email:
hisakato@sakura.cc.tsukuba.ac.jp

**Xiangdong Ye**

Affiliation:
Department of Mathematics, University of Science and Technology of China, Hefei, Anhui, 230026, People’s Republic of China

Email:
yexd@math.ustc.edu.cn

DOI:
https://doi.org/10.1090/S0002-9939-00-05247-3

Keywords:
Graph,
$n$-indecomposable,
$\epsilon$-map,
Burgess's theorem

Received by editor(s):
March 24, 1998

Received by editor(s) in revised form:
September 17, 1998

Published electronically:
February 25, 2000

Additional Notes:
This project was supported by NSFC 19625103 and JSPS of Japan.

Communicated by:
Alan Dow

Article copyright:
© Copyright 2000
American Mathematical Society