Dimension of the product and classical formulae of dimension theory

Abstract

Let $f : X \lo Y$ be a map of compact metric spaces. A classical theorem of Hurewicz asserts that $\dim X\le \dim Y+\dim f$ where $\dim f=\sup \{\dim f^{-1}(y):y\in Y\}$. The first author conjectured that {\em $\dim Y+\dim f$ in Hurewicz's theorem can be replaced by $\sup \{\dim (Y\times f^{-1}(y)):y\in Y\}$}. We disprove this conjecture. As a by-product of the machinery presented in the paper we answer in negative the following problem posed by the first author: {\em Can for compact $X$ the Menger-Urysohn formula $\dim X\le \dim A+\dim B+1$ be improved to $\dim X\le \dim (A\times B)+1$ ?} On a positive side we show that both conjectures holds true for compacta $X$ satisfying the equality $dim(X\times X)=2\dim X$.

References in corpus (1)

• Cohomological dimension theory of compact metric spaces