On dimensionally restricted maps
Volume 175 / 2002
Fundamenta Mathematicae 175 (2002), 35-52
MSC: Primary 54F45; Secondary 55M10, 54C65.
DOI: 10.4064/fm175-1-2
Abstract
Let $f : X\to Y$ be a closed $n$-dimensional surjective map of metrizable spaces. It is shown that if $Y$ is a $C$-space, then: (1) the set of all maps $g : X\to {\mathbb I}^n$ with $\mathop {\rm dim}\nolimits (f\mathbin {\triangle }g)=0$ is uniformly dense in $C(X,{\mathbb I}^n)$; (2) for every $0\leq k\leq n-1$ there exists an $F_{\sigma }$-subset $A_k$ of $X$ such that $\mathop {\rm dim}\nolimits A_k\leq k$ and the restriction $f|(X \setminus A_k)$ is $(n-k-1)$-dimensional. These are extensions of theorems by Pasynkov and Toruńczyk, respectively, obtained for finite-dimensional spaces. A generalization of a result due to Dranishnikov and Uspenskij about extensional dimension is also established.
