Beschreibung:
Suppose f : X â R n f:X \to {{\mathbf {R}}^n} is a continuous function from a closed subset X X of R n {{\mathbf {R}}^n} into R n {{\mathbf {R}}^n} . The Tietze Extension Theorem states that there is a continuous function F : R n â R n F:{{\mathbf {R}}^n} \to {{\mathbf {R}}^n} that extends f f . Here we consider the question of when the extension F F can be chosen with F | R n â X F|{{\mathbf {R}}^n} - X being finite-to-one. Not every map f f has such an extension. If f ( X ) f(X) is sufficiently nice, then there is such a finite-to-one extension. For example, it is shown that if f : X â R n f:X \to {{\mathbf {R}}^n} is a map and f ( X ) â R n â 1 Ă { 0 } f(X) \subset {{\mathbf {R}}^{n - 1}} \times \{ 0\} then there is a continuous extension F : R n â R n F:{{\mathbf {R}}^n} \to {{\mathbf {R}}^n} such that F | R n â X F|{{\mathbf {R}}^n} - X is finite-to-one. On the other hand, if X X is nowhere dense and f ( X ) 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 â R n f:X \to {{\mathbf {R}}^n} is not necessarily a function of the topology of f ( X ) f(X) , but may depend on its embedding or on the map f f .