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## On dimensionally restricted maps

### Tom 175 / 2002

Fundamenta Mathematicae 175 (2002), 35-52 MSC: Primary 54F45; Secondary 55M10, 54C65. DOI: 10.4064/fm175-1-2

#### Streszczenie

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.

#### Autorzy

• H. Murat TuncaliDepartment of Mathematics
Nipissing University
100 College Drive
P.O. Box 5002
North Bay, ON, P1B 8L7, Canada
e-mail
• Vesko ValovDepartment of Mathematics
Nipissing University
100 College Drive
P.O. Box 5002
North Bay, ON, P1B 8L7, Canada
e-mail

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