## Dimension-raising maps in a large scale

### Volume 223 / 2013

Fundamenta Mathematicae 223 (2013), 83-97
MSC: Primary 54F45; Secondary 54E35.
DOI: 10.4064/fm223-1-6

#### Abstract

Hurewicz's dimension-raising theorem states that $\dim Y \leq \dim X + n$ for every $n$-to-$1$ map $f: X\rightarrow Y$. In this paper we introduce a new notion of finite-to-one like map in a large scale setting. Using this notion we formulate a dimension-raising type theorem for asymptotic dimension and asymptotic Assouad–Nagata dimension. It is also well-known (Hurewicz's finite-to-one mapping theorem) that $\dim X \leq n$ if and only if there exists an $(n+1)$-to-$1$ map from a $0$-dimensional space onto $X$. We formulate a finite-to-one mapping type theorem for asymptotic dimension and asymptotic Assouad–Nagata dimension.