Product theorems in dimension theory
type: Item
instance of: scholarly article
title: Product theorems in dimension theory
zbMATH Open document ID: 0806.54028
published in: Tsukuba Journal of Mathematics
publication date: 1995-02-14T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 54F45
Mathematics Subject Classification ID: 54B10
zbMATH DE Number: 515572
zbMATH Keywords: \(F\)-product
zbMATH Keywords: piecewise rectangular product
author: B. A. Pasynkov
author: Koichi Tsuda
reviewed by: J. Keesling
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.21099/tkbjm/1496162129
OpenAlex ID: W2282640155
DOI: 10.21099/TKBJM/1496162129
description: scientific article; zbMATH DE number 515572