Maps with dimensionally restricted fibers

Vesko Valov¹

¹ Department of Computer Science and Mathematics Nipissing University 100 College Drive P.O. Box 5002 North Bay, ON, P1B 8L7, Canada

Colloquium Mathematicum, Tome 123 (2011) no. 2, pp. 239-248

Résumé

We prove that if $f\colon X\to Y$ is a closed surjective map between metric spaces such that every fiber $f^{-1}(y)$ belongs to a class $\mathrm S$ of spaces, then there exists an $F_\sigma$-set $A\subset X$ such that $A\in\mathrm S$ and $\dim f^{-1}(y)\setminus A=0$ for all $y\in Y$. Here, $\mathrm S$ can be one of the following classes: (i) $\{M:\mathop{\rm e\text{-}dim}\nolimits M\leq K\}$ for some $CW$-complex $K$; (ii) $C$-spaces; (iii) weakly infinite-dimensional spaces. We also establish that if $\mathrm S=\{M:\dim M\leq n\}$, then $\dim f\bigtriangleup g\leq 0$ for almost all $g\in C(X,\mathbb I^{n+1})$.

Détail

DOI : 10.4064/cm123-2-8

Keywords: prove colon closed surjective map between metric spaces every fiber belongs class mathrm spaces there exists sigma set subset mathrm dim setminus here mathrm following classes mathop text dim nolimits leq cw complex nbsp nbsp c spaces iii weakly infinite dimensional spaces establish mathrm dim leq dim bigtriangleup leq almost mathbb