Maps in with finite-to-one extensions

Author:
Michael Starbird

Journal:
Proc. Amer. Math. Soc. **98** (1986), 317-323

MSC:
Primary 54C20

MathSciNet review:
854040

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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 .

**[HW]**Witold Hurewicz and Henry Wallman,*Dimension Theory*, Princeton Mathematical Series, v. 4, Princeton University Press, Princeton, N. J., 1941. MR**0006493**

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DOI:
https://doi.org/10.1090/S0002-9939-1986-0854040-1

Keywords:
Finite-to-one,
extension

Article copyright:
© Copyright 1986
American Mathematical Society