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On Finite-to-One Maps

Published online by Cambridge University Press:  20 November 2018

H. Murat Tuncali  and
Vesko Valov
H. Murat Tuncali
Affiliation:
Department of Mathematics, Nipissing University, 100 College Drive, P.O. Box 5002, North Bay, ON, P1B 8L7 e-mail: muratt@nipissingu.ca e-mail: veskov@nipissingu.ca
Vesko Valov
Affiliation:
Department of Mathematics, Nipissing University, 100 College Drive, P.O. Box 5002, North Bay, ON, P1B 8L7 e-mail: muratt@nipissingu.ca e-mail: veskov@nipissingu.ca
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Abstract

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Let $f:X\to Y$ be a $\sigma$-perfect $k$-dimensional surjective map of metrizable spaces such that dim $Y\le m$. It is shown that for every positive integer $p$ with $p\le m+k+1$ there exists a dense ${{G}_{\delta }}-\text{subset}\,\mathcal{H}\left( k,m,p \right)$ of $C\left( X,{{\mathbb{I}}^{k+p}} \right)$ with the source limitation topology such that each fiber of $f\Delta g$, $g\in \mathcal{H}\left( k,m,p \right)$, contains at most $\max \left\{ k+m-p+2,1 \right\}$ points. This result provides a proof the following conjectures of S. Bogatyi, V. Fedorchuk and J. van Mill. Let $f:X\to Y$ be a $k$-dimensional map between compact metric spaces with dim $Y\le m$. Then: (1) there exists a map $h:X\to {{\mathbb{I}}^{m+2k}}$ such that $f\Delta h:\,X\to Y\times {{\mathbb{I}}^{m+2k}}$ is 2-to-one provided $k\ge 1$; (2) there exists a map $h:X\to {{\mathbb{I}}^{m+k+1}}$ such that $f\Delta h:X\to Y\times {{\mathbb{I}}^{m+k+1}}$ is $\left( k+1 \right)$-to-one.

Keywords

54F4555M1054C65Finite-to-one mapsdimensionset-valued maps
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 48 , Issue 4 , 01 December 2005 , pp. 614 - 621
Copyright
Copyright © Canadian Mathematical Society 2005

References

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