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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Mappings between Euclidean spaces that are one-to-one over the image of a dense subset


Authors: Mladen Bestvina and John J. Walsh
Journal: Proc. Amer. Math. Soc. 91 (1984), 449-455
MSC: Primary 57N15; Secondary 54B15, 54C05, 54C10
MathSciNet review: 744647
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Abstract: Examples are constructed that include, for $ n > m \geqslant 2$, proper surjective maps $ f:{{\mathbf{R}}^n} \to {{\mathbf{R}}^m}$ that are one to one over the image of a dense subset $ D \subset {{\mathbf{R}}^n}$; i.e., the restriction of $ f$ yields a homeomorphism from $ {f^{ - 1}}f(D)$ onto $ f(D)$. A routine Baire Category argument establishes that, for any $ \sigma $-compact subset $ F \subset {{\mathbf{R}}^n}$ which $ f$ maps onto $ {{\mathbf{R}}^m}$, $ {\operatorname{Int}}(F) \ne \theta $ and, hence, $ \dim F = n$.


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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9939-1984-0744647-2
PII: S 0002-9939(1984)0744647-2
Keywords: Deficient points, dendrite, manifold, stable values, upper semicontinuous decomposition
Article copyright: © Copyright 1984 American Mathematical Society