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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Maps in $ {\bf R}\sp n$ with finite-to-one extensions


Author: Michael Starbird
Journal: Proc. Amer. Math. Soc. 98 (1986), 317-323
MSC: Primary 54C20
MathSciNet review: 854040
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Abstract: Suppose $ f:X \to {{\mathbf{R}}^n}$ is a continuous function from a closed subset $ X$ of $ {{\mathbf{R}}^n}$ into $ {{\mathbf{R}}^n}$. The Tietze Extension Theorem states that there is a continuous function $ F:{{\mathbf{R}}^n} \to {{\mathbf{R}}^n}$ that extends $ f$. Here we consider the question of when the extension $ F$ can be chosen with $ F\vert{{\mathbf{R}}^n} - X$ being finite-to-one. Not every map $ f$ has such an extension. If $ f(X)$ is sufficiently nice, then there is such a finite-to-one extension. For example, it is shown that if $ f:X \to {{\mathbf{R}}^n}$ is a map and $ f(X) \subset {{\mathbf{R}}^{n - 1}} \times \{ 0\} $ then there is a continuous extension $ F:{{\mathbf{R}}^n} \to {{\mathbf{R}}^n}$ such that $ F\vert{{\mathbf{R}}^n} - X$ is finite-to-one. On the other hand, if $ X$ is nowhere dense and $ f(X)$ contains an open set, then there definitely is not such a finite-to-one extension. Other examples and theorems show that the finite-to-one extendability of a map $ f:X \to {{\mathbf{R}}^n}$ is not necessarily a function of the topology of $ f(X)$, but may depend on its embedding or on the map $ f$.


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DOI: http://dx.doi.org/10.1090/S0002-9939-1986-0854040-1
PII: S 0002-9939(1986)0854040-1
Keywords: Finite-to-one, extension
Article copyright: © Copyright 1986 American Mathematical Society