Computation of inductive dimensions of product of compacta (Q465855)

There are several reasons to study dimension-like invariants: to get more information about spaces; to extend the class of spaces under investigation; to evaluate the classical dimensions, etc. M. Charalambous, A. Chigogidze, V. Filippov, J. Aarts, T. Nishiura, B. Pasynkov, S. Iliadis are a few to be mentioned in connection with these themes of research. The last works of V. V. Fedorchuk are connected with the problem of finding unifying approaches to the study of set-theoretical methods in the classical dimension theory and algebraic methods in the investigation of cohomological dimensions. In this paper the authors demonstrate how base dimension \(I\) can be used to compute the classical dimension Ind of products. \textit{B. A. Pasynkov} [Sov. Math., Dokl. 10, 1402--1406 (1969); translation from Dokl. Akad. Nauk SSSR 189, 254-257 (1969; Zbl 0205.52803)] established the finite-dimensionality of the product of two finite-dimensional compacta. \textit{V. V. Filippov} [Sov. Math., Dokl. 13, 250--254 (1972); translation from Dokl. Akad. Nauk SSSR 202, 1016-1019 (1972; Zbl 0243.54032)] constructed an example of compact spaces \(X\), \(Y\) such that NEWLINE\[NEWLINE{\mathrm{ Ind}}X = 1,\;{\mathrm{ Ind}}Y = 2\;{\mathrm{ and}} \;{\mathrm{ Ind}} X\times Y \geq {\mathrm{ ind} } X\times Y>3.NEWLINE\]NEWLINE By Pasynkov's estimation \({\mathrm{ Ind}}X \times Y\leq 7\), and \({\mathrm{ ind}}X \times Y\leq 4\), see \textit{V. A. Chatyrko} and \textit{K. L. Kozlov} [Commentat. Math. Univ. Carol. 41, No. 3, 597--603 (2000; Zbl 1038.54012)].NEWLINENEWLINEIn this very interesting paper the authors give the following result:NEWLINENEWLINE{ Theorem.} For compact spaces \(X\), \(Y\) in Filippov's example we have NEWLINE\[NEWLINE{\mathrm{ Ind}}X = 1, \;{\mathrm{ Ind}} Y = 2\;{\mathrm{ and}}\;{\mathrm{ Ind}} X \times Y = {\mathrm{ ind}}X \times Y = 4.NEWLINE\]