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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



On embeddings with locally nice cross-sections

Author: J. L. Bryant
Journal: Trans. Amer. Math. Soc. 155 (1971), 327-332
MSC: Primary 57.05
MathSciNet review: 0276983
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Abstract: A $ k$-dimensional compactum $ {X^k}$ in euclidean space $ {E^n}(n - k \geqq 3)$ is said to be locally nice in $ {E^n}$ if $ {E^n} - {X^k}$ is $ 1$-ULC. In this paper we prove a general theorem which implies, in particular, that $ {X^k}$ is locally nice in $ {E^n}$ if the intersection of $ {X^k}$ with each horizontal hyperplane of $ {E^n}$ is locally nice in the hyperplane. From known results we obtain immediately that a $ k$-dimensional polyhedron $ P$ in $ {E^n}$ ( $ n - k \geqq 3$ and $ n \geqq 5$) is tame in $ {E^n}$ if each $ ({E^{n - 1}} \times \{ w\} ) - P(w \in {E^1})$ is $ 1$-ULC. However, by strengthening our general theorem in the case $ n = 4$, we are able to prove this result for $ n = 4$ as well. For example, an arc $ A$ in $ {E^4}$ is tame if each horizontal cross-section of $ A$ is tame in the cross-sectional hyperplane (that is, lies in an arc that is tame in the hyperplane).

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Keywords: Locally nice embeddings, $ 1$-ULC subsets of $ {E^n}$, tame embeddings, embeddings with tame cross-sections, embeddings with locally nice cross-sections, topological embeddings of compacta, topological embeddings of polyhedra
Article copyright: © Copyright 1971 American Mathematical Society