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Homogeneous Suslinian Continua

  Published:2011-01-26
 Printed: Jun 2011
  • D. Daniel,
    Lamar University, Department of Mathematics, Beaumont, TX, U.S.A.
  • J. Nikiel,
    Opole University, Institute of Mathematics and Informatics, Opole, Poland
  • L. B. Treybig,
    Texas A&M University, Department of Mathematics, College Station, TX, U.S.A.
  • H. M. Tuncali,
    Nipissing University, Faculty of Arts and Sciences, North Bay, ON
  • E. D. Tymchatyn,
    University of Saskatchewan, Department of Mathematics, Saskatoon, SK
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Abstract

A continuum is said to be Suslinian if it does not contain uncountably many mutually exclusive non-degenerate subcontinua. Fitzpatrick and Lelek have shown that a metric Suslinian continuum $X$ has the property that the set of points at which $X$ is connected im kleinen is dense in $X$. We extend their result to Hausdorff Suslinian continua and obtain a number of corollaries. In particular, we prove that a homogeneous, non-degenerate, Suslinian continuum is a simple closed curve and that each separable, non-degenerate, homogenous, Suslinian continuum is metrizable.
Keywords: connected im kleinen, homogeneity, Suslinian, locally connected continuum connected im kleinen, homogeneity, Suslinian, locally connected continuum
MSC Classifications: 54F15, 54C05, 54F05, 54F50 show english descriptions Continua and generalizations
Continuous maps
Linearly ordered topological spaces, generalized ordered spaces, and partially ordered spaces [See also 06B30, 06F30]
Spaces of dimension $\leq 1$; curves, dendrites [See also 26A03]
54F15 - Continua and generalizations
54C05 - Continuous maps
54F05 - Linearly ordered topological spaces, generalized ordered spaces, and partially ordered spaces [See also 06B30, 06F30]
54F50 - Spaces of dimension $\leq 1$; curves, dendrites [See also 26A03]
 

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