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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Means on chainable continua


Author: Mirosław Sobolewski
Journal: Proc. Amer. Math. Soc. 136 (2008), 3701-3707
MSC (2000): Primary 54F15
Published electronically: May 15, 2008
MathSciNet review: 2415058
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Abstract: By a mean on a space $ X$ we understand a mapping $ \mu:X\times X\to X$ such that $ \mu (x,y)=\mu(y,x)$ and $ \mu(x,x)=x$ for $ x,y\in X$. A chainable continuum is a metric compact connected space which admits an $ \varepsilon$- mapping onto the interval $ [0,1]$ for every number $ \varepsilon >0$. We show that every chainable continuum that admits a mean is homeomorphic to the interval. In this way we answer a question by P. Bacon. We answer some other questions concerning means as well.


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Additional Information

Mirosław Sobolewski
Affiliation: Instytut Matematyki, Banacha 2, Warszawa 02-097, Poland
Email: msobol@mimuw.edu.pl

DOI: http://dx.doi.org/10.1090/S0002-9939-08-09414-8
PII: S 0002-9939(08)09414-8
Keywords: Continuum, chainable, mean
Received by editor(s): September 22, 2006
Received by editor(s) in revised form: August 14, 2007
Published electronically: May 15, 2008
Communicated by: Alexander N. Dranishnikov
Article copyright: © Copyright 2008 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.