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Embedding Circle-Like Continua in the Plane

Published online by Cambridge University Press:  20 November 2018

R. H. Bing
R. H. Bing*
Affiliation:
University of Wisconsin
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A finite sequence of open sets L1 L2, … , Ln is called a linear chain if each Li intersects only the L's adjacent to it in the sequence. The finite sequence is a circular chain if we also insist that the first and last links intersect each other. The 1-skeleton of the covering is an arc for a linear chain and a simple closed curve for a linear chain.

A compact metric continuum X is called snake-like if for each ∈ > 0, X can be covered by a linear chain of mesh less than ∈. Likewise, X is called circle-like if for each ∈ > 0, X can be irreducibly covered with a circular chain of mesh less than ∈. This definition is more restrictive than that given in (3, p. 210) for there a pseudo-arc is not circle-like but here it is. The present usage is in keeping with definitions of Burgess.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 14 , 1962 , pp. 113 - 128
Copyright
Copyright © Canadian Mathematical Society 1962

References

1. Bing, R. H., Snake-like continua, Duke Math. J., 18 (1951), 653–663.Google Scholar
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3. Bing, R. H., A simple dosed curve is the only homogeneous bounded plane continuum that contains an arc, Can. J. Math., 12 (1960), 209–230.Google Scholar
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5. Burgess, C. E., Himogeneous continua which are almost chainable, Can. J. Math., 13 (1961), 519–528.Google Scholar
6. Burgess, C. E., Homogeneous continua which are circularly chainable and contain an arc, Abstract 545-16, Amer. Math. Soc. Notices, 5 (1958), 229.Google Scholar
7. Roberts, R. H. and Steenrod, N. E., Monotone transformations of two-dimensional manifolds Ann. Math., 39 (1938), 851–863.Google Scholar