Maps in with finite-to-one extensions
Abstract: Suppose is a continuous function from a closed subset of into . The Tietze Extension Theorem states that there is a continuous function that extends . Here we consider the question of when the extension can be chosen with being finite-to-one. Not every map has such an extension. If is sufficiently nice, then there is such a finite-to-one extension. For example, it is shown that if is a map and then there is a continuous extension such that is finite-to-one. On the other hand, if is nowhere dense and 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 is not necessarily a function of the topology of , but may depend on its embedding or on the map .
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