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Transactions of the American Mathematical Society

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

 

 

Continua for which the set function $ T$ is continuous


Author: David P. Bellamy
Journal: Trans. Amer. Math. Soc. 151 (1970), 581-587
MSC: Primary 54.55
DOI: https://doi.org/10.1090/S0002-9947-1970-0271910-9
MathSciNet review: 0271910
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Abstract: The set-valued set function T has been studied extensively as an aid to classifying metric and Hausdorff continua. It is a consequence of earlier work of the author with H. S. Davis that T, considered as a map from the hyperspace of closed subsets of a compact Hausdorff space to itself, is upper semicontinuous. We show that in a continuum for which T is actually continuous (in the exponential, or Vietoris finite, topology) semilocal connectedness implies local connectedness, and raise the question of whether any nonlocally connected continuum for which T is continuous must be indecomposable.


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DOI: https://doi.org/10.1090/S0002-9947-1970-0271910-9
Keywords: Continuum neighborhoods, continuous set-valued functions, almost connected im kleinen, set function T, T-additive, T-symmetric, indecomposable continuum, semilocally connected continuum
Article copyright: © Copyright 1970 American Mathematical Society