DIMENSION OF THE PRODUCT AND CLASSICAL FORMULAE OF DIMENSION THEORY

Let f : X -> Y be a map of compact metric spaces. A classical theorem of Hurewicz asserts that dimX <= dim Y vertical bar dimf, where dim f = sup{dim f(-1) (y) : y is an element of Y}. The first author conjectured that dim Y + dimf in Hurewicz's theorem can be replaced by sup{dim(Y x f(-1)(y)) : y is an element of 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 dimX <= dimA + dimB + 1 for a decomposition of a compactum X = A boolean OR B into two sets be improved to the inequality dimX <= dim(A x B) + 1? On a positive side we show that both conjectures hold true for compacta X satisfying the equality dim(X x X) = 2dimX.