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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
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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Keywords: Flat spheres, wild spheres, crumpled <I>n</I>-cubes, cell unions, visible sets, seen sets, 1-ULC
Article copyright: © Copyright 1980 American Mathematical Society