Characterization of compact spaces with noncoinciding dimensions which are subsets of products of simple spaces
First examples of compact spaces with noncoinciding classical dimensions ind and \(\dim\) were realized as closed subsets of the product of the long closed interval \([0,\omega_c]\), where \(\omega_c\) is the first ordinal of cardinality of the continuum, and the closed bounded interval \([0,1]\). In the present paper the author studies dimensional properties of closed subsets of products \(\Pi=[0,\omega_c] \times K\), where \(K\) is a metrizable compactum. The first result is the following: for any closed subset \(X\) of \(\Pi\) we have \(\operatorname{Ind}X \leq \dim X +1.\) This implies the splitting of all closed subsets \(X\) of \(\Pi\) into two classes: either \(\operatorname{Ind}X= \operatorname{ind}X= \dim X +1\) or \(\dim X= \operatorname{ind}X \leq \operatorname{Ind} X\). The second (main) result is a characterization of closed subsets \(X\) of \(\Pi\) having \(\dim X = 1\) and \(\operatorname{ind}X= \operatorname{Ind}X =2\). Let us note that the existence of a closed subset \(X\) of \(\Pi\) with \(\dim X= \operatorname{ind}X< \operatorname{Ind}X\) is an open problem.
scientific article; zbMATH DE number 5365215