Covering dimension of topological products
This paper is dedicated to investigating the conditions under which the inequality \[ \dim X \times Y \leq \dim X + \dim Y\tag{+} \] holds. We already know that there are two compact metrizable 2-dimensional spaces such that their product is 3-dimensional from a result from L. S. Pontryagin. But generally we have that the inequality (+) is true. E.g., it holds for compact Hausdorff spaces and metrizable spaces. However we have an example of a product where one of the factors is metrizable and the other is Tychonoff. The question whether (+) holds for paracompact Hausdorff spaces is still open, also after this extensive research. (Question 3.25). Using the concept of rectangularity the authors are able to close in on this open question. So they prove (Theorem 3:15) that (+) holds for paracompact Hausdorff spaces \(X\) and \(Y\) where \(X\) is a \(\Sigma(\theta)\) space and \(Y\) is a \(P(\theta)\) space. This product is also paracompact. For conditions on one of the factors several results are strengthened. So if \(Y\) is \(\sigma\)-compact then (+) holds. Some results are proved when \(X\) is a metrizable space or the perfect image of a metrizable space. The other objective is to construct spaces \(X\) and \(Y\) such that \(\dim X \times Y > \dim X + \dim Y\). Various examples have been constructed in the past. Using spaces of consistent families the first author managed to construct new examples of spaces \(X\) and \(Y\) such that \(\dim X \times Y > \dim X + \dim Y\). He constructed a perfectly normal product by the Juhász-Kunen-Rudin method modified by Wage and corrected by Przymusinski. He also constructed normal products by methods suggested by van Douwen and developed by Przymusinski. The work is quite extensive and it gives an impression of the determination of the authors to strengthen results on the question of the product theorem beyond the field of metrizable spaces. It gives a good overview on the results proved on this issue and a basis for solving the main question whether for paracompact Hausdorff spaces the inequality (+) holds.
scientific article; zbMATH DE number 5319780
scientific article