Product theorems in dimension theory
The spaces in this paper are just assumed to be topological with no other assumptions unless specified. The following theorem is due to the first author. Theorem. Let \(X \times Y\) be piecewise rectangular. Then \(\text{Id} (X \times Y) \leq \text{Id} (X) + \text{Id} (Y)\). Here \(\text{Id} (Z)\) is a dimension function introduced by the first author in previous work. In this paper the authors prove extensions of this basic theorem by proving special cases of the following conjecture. Let \(\Pi = X_ 1' \times X_ 2\), \(* \in X_ 1'\), \(X_ 1 = X_ 1' \backslash \{*\}\), and the product \(\Pi_ 0 = X_ 1 \times X_ 2\) be piecewise rectangular and satisfy the following condition \((\#)\): Every set \(H\) is functionally separated from \(\{*\} \times X_ 2\) whenever \(H\) is closed in \(\Pi\) and \(H \cap (\{*\} \times X_ 2) = \emptyset\). Then, we have \(\text{Id} (\Pi) \leq \text{Id} (X_ 1') + \text{Id} (X_ 2)\). Applications and examples are given for the conditions under which the conjecture can be proved. It should be noted that the rectangular product is just the \(F\)- product of Nagata if one is dealing with normal spaces.
scientific article; zbMATH DE number 515572