Dimension of the product and classical formulae of dimension theory

TL;DR

This paper investigates the relationships between dimensions of compact metric spaces and maps, disproves a conjecture related to classical dimension theory, and explores conditions under which certain dimension formulas hold.

Contribution

It disproves a conjecture replacing the sum of dimensions with a supremum involving product spaces and examines when classical dimension formulas can be improved.

Findings

Disproved the conjecture that $ ext{dim }Y + ext{dim }f$ can be replaced by $ ext{sup} ext{dim }(Y imes f^{-1}(y))$.

Showed that the Menger-Urysohn formula cannot generally be improved to involve the dimension of the product.

Identified conditions, such as $ ext{dim}(X imes X) = 2 ext{dim }X$, under which the classical formulas hold true.

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 $dim(X\times X)=2\dim X$.