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 $\dim X \leq \dim Y +\dim f$ 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 $\dim X \leq \dim A + \dim B +1$ be improved to $\dim X \leq \dim (A \times B) +1$ ?} On a positive side we show that both conjectures holds true for compacta $X$ satisfying the equality $dim(X\times X)=2\dim X$.