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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

$ \omega $-connected continua and Jones' $ K$ function


Author: Eldon J. Vought
Journal: Proc. Amer. Math. Soc. 91 (1984), 633-636
MSC: Primary 54F20; Secondary 54B15, 54F65
MathSciNet review: 746104
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Abstract: A continuum $ X$ is $ \omega $-connected if for every pair of points $ x$, $ y$ of $ X$, there exists an irreducible subcontinuum of $ X$ from $ x$ to $ y$ that is decomposable. If $ A \subset X$ then $ K\left( A \right)$ is the intersection of all subcontinua of $ X$ that contain $ A$ in their interiors. The main theorem shows that if $ X$ is an $ \omega $-connected continuum and $ H$ is a connected nowhere dense subset of $ X$, then $ K\left( H \right)$ has a void interior. Several corollaries are established for continua with certain separation properties and a final theorem shows the equivalence of $ \omega $-connectedness and $ \delta $-connectedness for plane continua.


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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9939-1984-0746104-6
PII: S 0002-9939(1984)0746104-6
Keywords: $ \omega $-connectedness, $ \delta $-connectedness, $ \lambda $-connectedness, $ \alpha $-connectedness, Jones' $ K$ function, $ {\theta _n}$-continuum, $ \theta $-continuum, upper-semicontinuous decomposition, Knaster indecomposable continuum with one endpoint
Article copyright: © Copyright 1984 American Mathematical Society