Unions of cells with applications to visibility

Loveland, L. D.

doi:10.1090/S0002-9939-1980-0556636-8

Unions of cells with applications to visibility
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by L. D. Loveland

Proc. Amer. Math. Soc. 78 (1980), 580-584

DOI: https://doi.org/10.1090/S0002-9939-1980-0556636-8

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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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