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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 

 

Schlais' theorem extends to $ \lambda $ connected plane continua


Author: Charles L. Hagopian
Journal: Proc. Amer. Math. Soc. 40 (1973), 265-267
MSC: Primary 54F15; Secondary 54F20
DOI: https://doi.org/10.1090/S0002-9939-1973-0317297-2
MathSciNet review: 0317297
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Abstract: We call a nondegenerate metric space that is compact and connected a continuum. For each point $ x$ of a continuum $ M$, F. B. Jones [2] defines $ K(x)$ to be the set consisting of all points $ y$ of $ M$ such that $ M$ is not aposyndetic at $ x$ with respect to $ y$. H. E. Schlais [4] proved that if $ M$ is a hereditarily decomposable continuum, then for each point $ x$ of $ M$, no nonempty open set in $ M$ is contained in $ K(x)$. A continuum $ M$ is said to be $ \lambda $ connected if any two of its points can be joined by a hereditarily decomposable continuum in $ M$. Here we prove that if $ M$ is a $ \lambda $ connected plane continuum, then for each point $ x$ of $ M$, the set $ K(x)$ does not contain a nonempty open subset of $ M$.


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DOI: https://doi.org/10.1090/S0002-9939-1973-0317297-2
Keywords: Hereditarily decomposable continua, lambda connected plane continua, Jones functions $ K$, arcwise connectivity, Schlais theorem
Article copyright: © Copyright 1973 American Mathematical Society