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Homogeneous Suslinian Continua |
Canad. Math. Bull. 54(2011), 244-248
Published:2011-01-26
Printed: Jun 2011
Printed: Jun 2011
- D. Daniel,
- J. Nikiel,
- L. B. Treybig,
- H. M. Tuncali,
- E. D. Tymchatyn,
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 |
| MSC Classifications: |
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] |