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

  • R. H. Bing (a1)

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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.

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References

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1. Anderson, R.D., A characterization of the universal curve and a proof of its homogeneity, Ann. Math., 67 (1958), 313324.
2. Bing, R.H., A homogeneous indecomposable plane continuum, Duke Math., 16 (1948), 729742.
3. Bing, R.H., Snake-like continua, Duke Math. J., 18 (1951), 653663.
4. Bing, R.H., Concerning hereditarily indecomposable continua, Pac. J. Math., 1 (1951), 4351.
5. Bing, R.H., Each homogeneous chainable continuum is a pseudo arc, Proc. Amer. Math. Soc, 10 (1959), 345346.
6. Bing, R.H. and Jones, F.B., Another homogeneous plane continuum, Trans. Amer. Math. Soc, 90 (1959), 171192.
7. Burgess, C.E., Continua and various types of homogeneity, Trans. Amer. Math. Soc., 88 (1958), 366374.
8. Cohen, H.J., Some results concerning homogeneous plane continua, Duke Math. J., 18 (1951), 467474.
9. Grace, E.E., Totally nonconnected im kleinen continua, Proc. Amer. Math. Soc, 9 (1958), 818821.
10. Jones, F.B., Certain homogeneous unicoherent indecomposable continua, Proc. Amer. Math. Soc, 9 (1951), 855859.
11. Mazurkiewicz, S., Sur les continus homogènes, Fund. Math., 5 (1924), 137146.
12. Moise, E.E., A note on the pseudo-arc, Trans. Amer. Math. Soc, 64 (1949), 5758.
13. Moore, R.L., Foundations of point set theory, Amer. Math. Soc Coll. Publ., 13 (1932).
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A Simple Closed Curve is the Only Homogeneous Bounded Plane Continuum that Contains an Arc

  • R. H. Bing (a1)

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