Dimension of the product and classical formulae of dimension theory

Let f: X→y Y be a map of compact metric spaces. A classical theorem of Hurewicz asserts that dim X ≤ dim Y + dim f, where dim f = sup{dim f^-1(y): y ε Y}. The first author conjectured that dim Y + dim f in Hurewicz's theorem can be replaced by sup{dim(Y × f^-1(y)): y ε 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 ≤ dim A + dimB + 1 for a decomposition of a compactum X = A ∪ B into two sets be improved to the inequality dim X ≤ dim(A × B) + 1?. On a positive side we show that both conjectures hold true for compacta X satisfying the equality dim(X × X) = 2 dim X.