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A Simple Closed Curve is the Only Homogeneous Bounded Plane Continuum that Contains an Arc

Published online by Cambridge University Press:  20 November 2018

R. H. Bing
R. H. Bing*
Affiliation:
University of Wisconsin
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One of the unsolved problems of plane topology is the following:

Question. What are the homogeneous bounded plane continua?

A search for the answer has been punctuated by some erroneous results. For a history of the problem see (6).

The following examples of bounded homogeneous plane continua are known : a point; a simple closed curve; a pseudo arc (2, 12); and a circle of pseudo arcs (6). Are there others?

The only one of the above examples that contains an arc is a simple closed curve. In this paper we show that there are no other such examples. We list some previous results that point in this direction. Mazurkiewicz showed (11) that the simple closed curve is the only non-degenerate homogeneous bounded plane continuum that is locally connected. Cohen showed (8) that the simple closed curve is the only homogeneous bounded plane continuum that contains a simple closed curve.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 12 , 1960 , pp. 209 - 230
Copyright
Copyright © Canadian Mathematical Society 1960

References

1. Anderson, R.D., A characterization of the universal curve and a proof of its homogeneity, Ann. Math., 67 (1958), 313–324.Google Scholar
2. Bing, R.H., A homogeneous indecomposable plane continuum, Duke Math., 16 (1948), 729–742.Google Scholar
3. Bing, R.H., Snake-like continua, Duke Math. J., 18 (1951), 653–663.Google Scholar
4. Bing, R.H., Concerning hereditarily indecomposable continua, Pac. J. Math., 1 (1951), 43–51.Google Scholar
5. Bing, R.H., Each homogeneous chainable continuum is a pseudo arc, Proc. Amer. Math. Soc, 10 (1959), 345–346.Google Scholar
6. Bing, R.H. and Jones, F.B., Another homogeneous plane continuum, Trans. Amer. Math. Soc, 90 (1959), 171–192.Google Scholar
7. Burgess, C.E., Continua and various types of homogeneity, Trans. Amer. Math. Soc., 88 (1958), 366–374.Google Scholar
8. Cohen, H.J., Some results concerning homogeneous plane continua, Duke Math. J., 18 (1951), 467–474.Google Scholar
9. Grace, E.E., Totally nonconnected im kleinen continua, Proc. Amer. Math. Soc, 9 (1958), 818–821.Google Scholar
10. Jones, F.B., Certain homogeneous unicoherent indecomposable continua, Proc. Amer. Math. Soc, 9 (1951), 855–859.Google Scholar
11. Mazurkiewicz, S., Sur les continus homogènes, Fund. Math., 5 (1924), 137–146.Google Scholar
12. Moise, E.E., A note on the pseudo-arc, Trans. Amer. Math. Soc, 64 (1949), 57–58.Google Scholar
13. Moore, R.L., Foundations of point set theory, Amer. Math. Soc Coll. Publ., 13 (1932).Google Scholar