Dimension of the product and classical formulae of dimension theory

Abstract:

Let $f : X \longrightarrow Y$ 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 $\dim Y + \dim f$ 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 the negative the following problem posed by the first author: Can the Menger-Urysohn Formula $\dim X \leq \dim A + \dim B +1$ for a decomposition of a compactum $X=A\cup B$ into two sets be improved to the inequality $\dim X \leq \dim (A \times B) +1$?

On a positive side we show that both conjectures hold true for compacta $X$ satisfying the equality $\dim (X\times X)=2\dim X$.