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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Homogeneous continua in Euclidean $ (n+1)$-space which contain an $ n$-cube are $ n$-manifolds


Author: Janusz R. Prajs
Journal: Trans. Amer. Math. Soc. 318 (1990), 143-148
MSC: Primary 54F20; Secondary 57N35
MathSciNet review: 943307
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Abstract: Let $ X$ be a homogeneous continuum and let $ {E^n}$ be Euclidean $ n$-space. We prove that if $ X$ is properly contained in a connected $ (n + 1)$-manifold, then $ X$ contains no $ n$-dimensional umbrella (i.e. a set homeomorphic to the set $ \{ ({x_1}, \ldots ,{x_{n + 1}}) \in {E^{n + 1}}:x_1^2 + \cdots + x_{n + 1}^2 \leq 1$ and $ {x_{n + 1}} \leq 0$ and either $ {x_1} = \cdots = {x_n} = 0$ or $ {x_{n + 1}} = 0\} $). Combining this fact with an earlier result of the author we conclude that if $ X$ lies in $ {E^{n + 1}}$ and topologically contains $ {E^n}$, then $ X$ is an $ n$-manifold.


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DOI: http://dx.doi.org/10.1090/S0002-9947-1990-0943307-X
PII: S 0002-9947(1990)0943307-X
Keywords: Continuum, Euclidean space, homogeneity, $ n$-dimensional umbrella, $ n$-manifold
Article copyright: © Copyright 1990 American Mathematical Society