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

 

Relating spans of some continua homeomorphic to $ S\sp n$


Author: Thelma West
Journal: Proc. Amer. Math. Soc. 112 (1991), 1185-1191
MSC: Primary 54F15
MathSciNet review: 1070534
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Abstract: It has been conjectured that $ {\sigma ^ * }(X) \geq \tfrac{1}{2}\sigma _0^ * (X)$ for each nonempty connected metric space $ X$. In this paper we show that $ {\sigma ^ * }(X) \geq (\sqrt 3 /2)\sigma _0^ * (X)$ when $ X \subset {R^n}$ is homeomorphic to $ {S^{n - 1}}$ for $ n = 2,3, \ldots $ and $ A$ is convex where $ A$ is the bounded component of $ {R^n} - X$. We also show that under certain conditions a lower bound for the ratio $ {\sigma ^ * }(X)/\sigma _0^ * (X)$ is larger than $ \sqrt 3 /2$. It has also been conjectured that $ {\sigma ^ * }(X) \geq \sigma (X)/2$ and that $ \sigma _0^ * (X) \geq {\sigma _0}(X)/2$ for each nonempty connected metric space $ X$. We show that these two inequalities hold when $ X \subset {R^n}$ is homeomorphic to $ {S^{n - 1}}$ for $ n = 3,4, \ldots $ and $ A$ is convex where $ A$ is the bounded component of $ {R^n} - X$.


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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9939-1991-1070534-2
PII: S 0002-9939(1991)1070534-2
Keywords: Span, convex continua
Article copyright: © Copyright 1991 American Mathematical Society



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