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Medientyp:
E-Artikel
Titel:
Maps in Rnwith Finite-to-One Extensions
Beteiligte:
Starbird, Michael
Erschienen:
American Mathematical Society, 1986
Erschienen in:
Proceedings of the American Mathematical Society, 98 (1986) 2, Seite 317-323
Sprache:
Englisch
ISSN:
0002-9939;
1088-6826
Entstehung:
Anmerkungen:
Beschreibung:
Suppose f: X → Rnis a continuous function from a closed subset X of Rninto Rn. The Tietze Extension Theorem states that there is a continuous function F: Rn→ Rnthat extends f. Here we consider the question of when the extension F can be chosen with F∣Rn- 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 → Rnis a map and$f(X) \subset \mathbf{R}^{n - 1} \times \{0 \}$then there is a continuous extension F: Rn→ Rnsuch that F∣Rn- 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 → Rnis not necessarily a function of the topology of f(X), but may depend on its embedding or on the map f.