# Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

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

The 2020 MCQ for Transactions of the American Mathematical Society is 1.48 .

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## Homogeneous continua in Euclidean $(n+1)$-space which contain an $n$-cube are $n$-manifoldsHTML articles powered by AMS MathViewer

by Janusz R. Prajs
Trans. Amer. Math. Soc. 318 (1990), 143-148 Request permission

## 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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