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

Published electronically:
February 25, 2000

MathSciNet review:
1653402

Full-text PDF Free Access

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**0054244**, 10.1090/S0002-9939-1953-0054244-2**[B2]**C. E. Burgess,*Chainable continua and indecomposability*, Pacific J. Math.**9**(1959), 653–659. MR**0110999****[K]**Hisao Kato,*A note on refinable maps and quasihomeomorphic compacta*, Proc. Japan Acad. Ser. A Math. Sci.**58**(1982), no. 2, 69–71. MR**651293****[LXY]**J. Lu, J. Xiong and X. Ye, The inverse limit space and the dynamics of a graph map, Preprint, 1997.**[MS]**Sibe Mardešić and Jack Segal,*𝜖-mappings onto polyhedra*, Trans. Amer. Math. Soc.**109**(1963), 146–164. MR**0158367**, 10.1090/S0002-9947-1963-0158367-X**[N]**Sam B. Nadler Jr.,*Continuum theory*, Monographs and Textbooks in Pure and Applied Mathematics, vol. 158, Marcel Dekker, Inc., New York, 1992. An introduction. MR**1192552**

Retrieve articles in *Proceedings of the American Mathematical Society*
with MSC (1991):
54B15,
54F15,
54F50

Retrieve articles in all journals with MSC (1991): 54B15, 54F15, 54F50

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