## Unions of cells with applications to visibility

HTML articles powered by AMS MathViewer

- by L. D. Loveland
- Proc. Amer. Math. Soc.
**78**(1980), 580-584 - DOI: https://doi.org/10.1090/S0002-9939-1980-0556636-8
- PDF | 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

- W. R. Alford,
*Some βniceβ wild $2$-spheres in $E^{3}$*, Topology of 3-manifolds and related topics (Proc. The Univ. of Georgia Institute, 1961) Prentice-Hall, Englewood Cliffs, N.J., 1962, pp.Β 29β33. MR**0141091** - R. H. Bing,
*A surface is tame if its complement is $1$-ULC*, Trans. Amer. Math. Soc.**101**(1961), 294β305. MR**131265**, DOI 10.1090/S0002-9947-1961-0131265-1 - C. E. Burgess and J. W. Cannon,
*Embeddings of surfaces in $E^{3}$*, Rocky Mountain J. Math.**1**(1971), no.Β 2, 259β344. MR**278277**, DOI 10.1216/RMJ-1971-1-2-259 - James W. Cannon,
*$^{\ast }$-taming sets for crumpled cubes. I. Basic properties*, Trans. Amer. Math. Soc.**161**(1971), 429β440. MR**282353**, DOI 10.1090/S0002-9947-1971-0282353-7 - J. W. Cannon,
*$\textrm {ULC}$ properties in neighbourhoods of embedded surfaces and curves in $E^{3}$*, Canadian J. Math.**25**(1973), 31β73. MR**314037**, DOI 10.4153/CJM-1973-004-1 - Robert J. Daverman,
*Embeddings of $(n-1)$-spheres in Euclidean $n$-space*, Bull. Amer. Math. Soc.**84**(1978), no.Β 3, 377β405. MR**645404**, DOI 10.1090/S0002-9904-1978-14476-0 - L. D. Loveland,
*Crumpled cubes that are finite unions of cells*, Houston J. Math.**4**(1978), no.Β 2, 223β228. MR**500856** - Carl P. Pixley,
*On crumpled cubes in $S^{3}$ which are the finite union of tame $3$-cells*, Houston J. Math.**4**(1978), no.Β 1, 105β112. MR**467756**

## Bibliographic 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