Maps in $\textbf {R}^ n$ with finite-to-one extensions
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- by Michael Starbird PDF
- Proc. Amer. Math. Soc. 98 (1986), 317-323 Request permission
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 $F|{{\mathbf {R}}^n} - X$ 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 $F|{{\mathbf {R}}^n} - X$ 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 $f$.References
- Witold Hurewicz and Henry Wallman, Dimension Theory, Princeton Mathematical Series, vol. 4, Princeton University Press, Princeton, N. J., 1941. MR 0006493
Additional Information
- © Copyright 1986 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 98 (1986), 317-323
- MSC: Primary 54C20
- DOI: https://doi.org/10.1090/S0002-9939-1986-0854040-1
- MathSciNet review: 854040