On the finite-dimensionality of topological products
This is an important paper in (at least) dimension theory. In this paper a complete and well-prepared proof (i.e. we believe that any standard textbook concerning dimension theory can accept it as a wonderful proof of this beautiful theorem) is given for one of his well-known announced results in [Sov. Math., Dokl. 10, 1402-1406 (1969); translation from Dokl. Akad. Nauk SSSR 189, 254-257 (1969; Zbl 0205.52803)]. In particular, a detailed proof of the following (nearly 30 years old) beautiful theorem is given here for the first time: If \(X\) is paracompact, \(Y\) is locally compact and \(\text{Ind }X, \text{Ind }Y <\infty\), then \(\text{Ind }X\times Y <\infty\). Moreover, there exist integers \(d(k,l)\leq -1\) for \(k,l = -1,0,1,\dots\) such that \(\text{Ind } X\times Y\geq d(\text{Ind } X, \text{Ind }Y)\). Moreover, the author gives here a proof for the more general case of (not necessarily paracompact) normal product space \(X\times Y\) in Theorem 1.1. Let the reviewer remind the reader that there still remain lots of good results due to the author, whose proofs have not been published yet (e.g. the reader can see many of them in a survey article [\textit{B. A. Pasynkov, V. V. Fedorchuk} and \textit{V. V. Filippov}, J. Sov. Math. 18, 789-841 (1982); translation from Itogi Nauki Tekh., Ser. Algebra Topologiya Geom. 17, 229-306 (1979; Zbl 0444.54024)]). Let me expect the author, one of the authorities in dimension theory, to publish detailed proofs for them, since no one (as far as I know) has reproved them over 20 years!
scientific article; zbMATH DE number 1121474
scientific article