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Homogeneous Suslinian Continua

Published online by Cambridge University Press:  20 November 2018

D. Daniel ,
J. Nikiel ,
L. B. Treybig ,
H. M. Tuncali  and
E. D. Tymchatyn
D. Daniel
Affiliation:
Lamar University, Department of Mathematics, Beaumont, TX, U.S.A.e-mail: dale.daniel@lamar.edu
J. Nikiel
Affiliation:
Opole University, Institute of Mathematics and Informatics, Opole, Polande-mail: nikiel@math.uni.opole.pl
L. B. Treybig
Affiliation:
Texas A&M University, Department of Mathematics, College Station, TX, U.S.A.e-mail: muratt@nipissingu.ca
H. M. Tuncali
Affiliation:
Nipissing University, Faculty of Arts and Sciences, North Bay, ONe-mail: tymchat@snoopy.usask.ca
E. D. Tymchatyn
Affiliation:
University of Saskatchewan, Department of Mathematics, Saskatoon, SK
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Abstract

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A continuumis 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

54F1554C0554F0554F50connected im kleinenhomogeneitySuslinianlocally connected continuum
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 54 , Issue 2 , 01 June 2011 , pp. 244 - 248
Copyright
Copyright © Canadian Mathematical Society 2011

References

[1] Banakh, T., Fedorchuk, V. V., Nikiel, J., and Tuncali, M., The Suslinian number and other cardinal invariants of continua. Fund. Math. 209(2010), no. 1, 43–57. doi:10.4064/fm209-1-4Google Scholar
[2] Cook, H. and Lelek, A., On the topology of curves. IV. Fund. Math. 76(1972), no. 2, 167–179.Google Scholar
[3] Daniel, D., Nikiel, J., Treybig, L. B., Tuncali, H. M., and Tymchatyn, E. D., On Suslinian continua. Canad. Math. Bull. 48(2005), no. 2, 195–202.Google Scholar
[4] Fitzpatrick, B. Jr. and Lelek, A., Some local properties of Suslinian compacta. Colloq. Math. 31(1974), 189–197.Google Scholar
[5] Gordh, G. R. Jr., Monotone decompositions of irreducible Hausdorff continua. Pacific J. Math. 36(1971), 647–658.Google Scholar
[6] Lelek, A., On the topology of curves. II. Fund. Math. 70(1971), no. 2, 131–138.Google Scholar
[7] Lelek, A. and Mohler, L., On the topology of curves. III. Fund. Math. 71(1971), no. 2, 147–160.Google Scholar
[8] Nikiel, J., The Hahn-Mazurkiewicz theorem for hereditarily locally connected continua. Topology Appl. 32(1989), no. 3, 307–323. doi:10.1016/0166-8641(89)90037-0Google Scholar
[9] Nikiel, J. and Tymchatyn, E. D., On homogeneous images of compact ordered spaces. Canad. J. Math. 45(1993), no. 2, 380–393.Google Scholar
[10] Pearson, B. J., Mapping arcs and dendritic spaces onto netlike continua. Colloq. Math. 34(1975/76), no. 1, 39–48.Google Scholar
[11] Simone, J., Concerning hereditarily locally connected continua. Colloq. Math. 39(1978), no. 2, 243–251.Google Scholar
[12] Treybig, L. B., Concerning continua which are continuous images of compact ordered spaces. Duke Math. J. 32(1965), 417–422. doi:10.1215/S0012-7094-65-03241-2Google Scholar
[13] Tymchatyn, E. D., Some n-arc theorems. Pacific J. Math. 66(1976), no. 1, 291–294.Google Scholar
[14] van Mill, J., On the character and π-weight of homogeneous compacta. Israel J. Math. 133(2003), 321–338. doi:10.1007/BF02773072Google Scholar
[15] Ward, L. E. Jr., The Hahn-Mazurkiewicz theorem for rim-finite continua. General Topology and Appl. 6(1976), no. 2, 183–190. doi:10.1016/0016-660X(76)90031-3Google Scholar
[16] Whyburn, G. T., Analytic topology. American Mathematical Society Colloquium Publications, 28, American Mathematical Society, New York, 1942.Google Scholar