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ArticleOriginal scientific text

Title

Continua which admit no mean

Authors 1, 2

Affiliations

  1. Institute of Mathematics, University of Tsukuba, Tsukuba-City, Ibaraki 305, Japan
  2. Department of Mathematics and Statistics, University of Saskatchewan, Saskatoon, Saskatchewan S7N 5E6, Canada

Abstract

A symmetric, idempotent, continuous binary operation on a space is called a mean. In this paper, we provide a criterion for the non-existence of mean on a certain class of continua which includes tree-like continua. This generalizes a result of Bell and Watson. We also prove that any hereditarily indecomposable circle-like continuum admits no mean.

Keywords

ENG
mean, pseudo-arc

Bibliography

  1. [Ba] P. Bacon, An acyclic continuum that admits no mean, Fund. Math. 67 (1970), 11-13.
  2. [BeW] M. Bell and S. Watson, Not all dendroids have means, preprint.
  3. [Bi1] R. H. Bing, A homogeneous indecomposable plane continuum, Duke Math. J. 15 (1948), 729-742.
  4. [Bi2] R. H. Bing, Snake-like continua, ibid. 18 (1951), 653-663.
  5. [C] D. W. Curtis, A hyperspace retraction theorem for a class of half-line compactifications, Topology Proc. 11 (1986), 29-64.
  6. [L] W. Lewis, Observations of the pseudo-arcs, ibid. 9 (1984), 329-337.
  7. [M] J. Mioduszewski, On a quasi-ordering in the class of continuous mappings of a closed interval, Colloq. Math. 9 (1962), 233-240.
  8. [O] L. G. Oversteegen, On products of confluent and weakly confluent mappings related to span, Houston J. Math. 12 (1986), 109-116.
  9. [S] K. Sigmon, Acyclicity of compact means, Michigan Math. J. 16 (1969), 111-115.
Colloquium Mathematicum
1996/71/1
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Pages:
97-105
Main language of publication
English
Received
1995-04-05
Accepted
1995-10-24
Published
1996
Exact and natural sciences