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Proceedings of the American Mathematical Society

Published by the American Mathematical Society, the Proceedings of the American Mathematical Society (PROC) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

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

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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Unions of cells with applications to visibility
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by L. D. Loveland PDF
Proc. Amer. Math. Soc. 78 (1980), 580-584 Request permission

Abstract:

A crumpled n-cell C in ${E^n}$ is proven to be an n-cell $(n \ne 4)$ when it is known to contain two n-cells ${C_1}$ and ${C_2}$, one of which is flat, such that ${\text {Bd}}\;C \subset ({\text {Bd}}\;{C_1}) \cup {\text {(Bd}}\;{C_2})$. This theorem is applied to show that C is an n-cell if its boundary is the union of two closed sets each of which is seen from some point of $\operatorname {Int} C$. Examples are given to show that flatness of one of ${C_1}$ and ${C_2}$ is necessary in the first theorem and to show that two is the largest integer for which either theorem is true.
References
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Additional Information
  • © Copyright 1980 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 78 (1980), 580-584
  • MSC: Primary 57N45
  • DOI: https://doi.org/10.1090/S0002-9939-1980-0556636-8
  • MathSciNet review: 556636