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A Characterization of Chainable Continua

Published online by Cambridge University Press:  20 November 2018

J. B. Fugate
J. B. Fugate*
Affiliation:
University of Kentucky, Lexington, Kentucky
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In this paper, certain results of Bing (1) and myself (2) are extended. It is well-known that a chainable compact metric continuum must be a-triodic (contain no triods), hereditarily unicoherent (the common part of each two subcontinua is connected), and each subcontinuum must be chainable. Our principal result states that a compact metric continuum M is chainable if and only if M is a-triodic, hereditarily unicoherent and each indecomposable subcontinuum of M is chainable. Some condition on the indecomposable subcontinua of M seems essential, if we consider the dyadic solenoid, 5, which is indecomposable, a-triodic and hereditarily unicoherent. Indeed, each proper subcontinuum of S is an arc. However, S is not chainable, since it cannot be embedded in the plane.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 21 , 1969 , pp. 383 - 393
Copyright
Copyright © Canadian Mathematical Society 1969

References

1. Bing, R. H., Snake-like continua, Duke Math. J. 19 (1951), 653–663.Google Scholar
2. Fugate, J. B., Decomposable chainable continua, Trans. Amer. Math. Soc. 123 (1966), 460–468.Google Scholar
3. Sorgenfrey, R. H., Concerning triodic continua, Amer. J. Math. 66 (1944), 439–460.Google Scholar