Maps in $\textbf {R}^ n$ with finite-to-one extensions

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 $f$.