The dimension problem for the product of infinite-dimensional spaces
Hurewicz introduced the concept of countable dimensionality and proved that the Hilbert cube was not countable dimensional. Alexandrov defined weakly infinite-dimensional spaces and asked whether every weakly infinite-dimensional space was countable dimensional. This famous problem became known as the Alexandrov problem. It was solved by \textit{R. Pol} in 1981 [Proc. Am. Math. Soc. 82, 634-636 (1981; Zbl 0469.54014)] who gave an example of a separable metric space which was weakly infinite-dimensional, but not countable-dimensional. There is another related problem which remains still to be solved. It is whether the product \(X \times Y\) of two weakly infinite-dimensional spaces \(X\) and \(Y\) is also weakly infinite-dimensional. This has become known as the product problem for infinite-dimensional spaces. The problem is unsolved, but in the past ten years considerable progress has been made. This paper is a summary of the research that has been done. The spaces studied in this problem are separable metric. The work has been done in several different directions. These form the four sections of the paper. The first section deals with classification of infinite-dimensional spaces by two-fiber and multifiber classifications. The second section deals with new results about the dimension of products of finite-dimensional spaces. The third section deals with the main concern of the survey, the problem of dimensionality for the product of infinite-dimensional spaces. The last section presents a variant of the definition of product, ordinal product, which is used to investigate the product problem for infinite-dimensional spaces. The survey not only summarizes current research, it also gives many classical theorems so that the reader is presented with a broad view of the history of this area. In addition to results, there is also a large collection of problems presented which are conveniently categorized. There is a large list of references of one hundred items.
scientific article; zbMATH DE number 907036
scientific article