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Homogeneous Continua Which are Almost Chainable1

Published online by Cambridge University Press:  20 November 2018

C. E. Burgess
C. E. Burgess*
Affiliation:
University of Utah
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The only known examples of nondegenerate homogeneous plane continua are the simple closed curve, the circle of pseudo-arcs (6), and the pseudo-arc (1; 13). Another example, called the pseudo-circle, has been suggested by Bing (2), but it has not been proved to be homogeneous. (Definitions of some of these terms and a history of results on homogeneous plane continua can be found in (6).) Of the three known examples, the pseudo-arc is both linearly chainable and circularly chainable, and the simple closed curve and the circle of pseudo-arcs are circularly chainable but not linearly chainable. It is not known whether every homogeneous plane continuum is either linearly chainable or circularly chainable. Bing has shown that a homogeneous continuum is a pseudo-arc provided it is linearly chainable (4).

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 13 , 1961 , pp. 519 - 528
Copyright
Copyright © Canadian Mathematical Society 1961

References

1. Bing, R. H., A homogeneous indecomposable plane continuum, Duke Math. J., 15 (1948), 729–742.Google Scholar
2. Bing, R. H., Concerning hereditarily indecomposable continua, Pacific J. Math., 1 (1951), 43–51.Google Scholar
3. Bing, R. H., Snake-like continua, Duke Math. J., 18 (1951), 653–663.Google Scholar
4. Bing, R. H., Each homogeneous nondegenerate chainable continuum is a pseudo-arc, Proc. Amer. Math. Soc, 10 (1959), 345–346.Google Scholar
5. Bing, R. H., A simple closed curve is the only homogeneous bounded plane continuum that contains an arc, Can. J. Math., 12 (1960), 209–230.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., Some theorems on n-homogeneous continua, Proc. Amer. Math. Soc., 5 (1954), 136–143.Google Scholar
8. Burgess, C. E., Homogeneous continua, Summary of Lectures and Seminars, Summer Institute on Set Theoretic Topology (Madison, 1955), 73–76.Google Scholar
9. Burgess, C. E., Continua and various types of homogeneity, Trans. Amer. Math. Soc, 88 (1958), 366–374.Google Scholar
10. Burgess, C. E., Chainable continua and indecomposability, Pacific J. Math., 9 (1959), 653–659.Google Scholar
11. Jones, F. B., Certain homogeneous unicoherent indecomposable continua, Proc. Amer. Math. Soc, 2 (1951), 855–859.Google Scholar
12. Moise, E. E., An indecomposable plane continuum which is homeomorphic to each of its nondegenerate subcontinua, Trans. Amer. Math. Soc, 63 (1948), 581–594.Google Scholar
13. Moise, E. E., A note on the pseudo-arc, Trans. Amer. Math. Soc, 64 (1949), 57–58.Google Scholar
14. Moore, R. L., Foundations of point set theory, Amer. Math. Soc Colloquium Publications, 13, 1932.Google Scholar
15. Rosen, Ronald H., On tree-like continua and irreducibility, Duke Math. J., 26 (1959), 113–122.Google Scholar
16. Sorgenfrey, R. H., Concerning triodic continua, Amer. J. Math., 66 (1944), 439–460.Google Scholar