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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 

 

$ \lambda $ connected plane continua


Author: Charles L. Hagopian
Journal: Trans. Amer. Math. Soc. 191 (1974), 277-287
MSC: Primary 54F20
DOI: https://doi.org/10.1090/S0002-9947-1974-0341435-4
MathSciNet review: 0341435
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Abstract: A continuum M is said to be $ {\mathbf{\lambda }}$ connected if any two distinct points of M can be joined by a hereditarily decomposable continuum in M. Recently this generalization of arcwise connectivity has been related to fixed point ptoblems in the plane. In particular, it is known that every $ {\mathbf{\lambda }}$ connected nonseparating plane continuum has the fixed point property. The importance of arcwise connectivity is, to a considerable extent, due to the fact that it is a continuous invariant. To show that $ {\mathbf{\lambda }}$ connectivity has a similar feature is the primary purpose of this paper. Here it is proved that if M is a $ {\mathbf{\lambda }}$ connected continuum and f is a continuous function of M into the plane, then $ f(M)$ is $ {\mathbf{\lambda }}$ connected. It is also proved that every semiaposyndetic plane continuum is $ {\mathbf{\lambda }}$ connected.


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DOI: https://doi.org/10.1090/S0002-9947-1974-0341435-4
Keywords: Arcwise connectivity, $ {\mathbf{\lambda }}$ connected continua, aposyndesis, semiaposyndesis, planar continuous images of hereditarily decomposable continua
Article copyright: © Copyright 1974 American Mathematical Society