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- W2029145248 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 $f$." @default.
- W2029145248 created "2016-06-24" @default.
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- W2029145248 date "1986-02-01" @default.
- W2029145248 modified "2023-09-25" @default.
- W2029145248 title "Maps in ${bf R}sp n$ with finite-to-one extensions" @default.
- W2029145248 doi "https://doi.org/10.1090/s0002-9939-1986-0854040-1" @default.
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