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.