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Proceedings of the American Mathematical Society

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

 

 

Unions of cells with applications to visibility


Author: L. D. Loveland
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
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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.


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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1980-0556636-8
Keywords: Flat spheres, wild spheres, crumpled n-cubes, cell unions, visible sets, seen sets, 1-ULC
Article copyright: © Copyright 1980 American Mathematical Society