On the estimation of the large inductive dimension of a product of compacta
The purpose of the paper under review is to prove that the large inductive dimension \(\text{Ind}(X_1\times X_2)\) of the product of compacta \(X_1\) and \(X_2\) with finite large inductive dimension is finite. The authors consider the base dimension \(\text{I}(X, {\mathcal F})\) for any pairs \((X, {\mathcal F})\) of Tychonoff spaces \(X\) and normal bases \({\mathcal F}\), which was defined in [\textit{D. Georgiou} et al., Mosc. Univ. Math. Bull. 64, No. 3, 95--101 (2009; Zbl 1304.54058); translation from Vest. Mosk. Univ. Mat. Mekh. 64, No. 3, 7--14 (2009)]. Here, the notion of normal base is in the sense of [\textit{O. Frink}, Am. J. Math. 86, 602--607 (1964; Zbl 0129.38101)]. The main result states that for any normal bases \({\mathcal F}_j\) on \(X_j\), \(j = 1, 2\), if \(\text{I}(X_j, {\mathcal F}_j) < \infty\) for \(j=1, 2\), then \(\text{I}(X_1\times X_2, {\mathcal F}_1 \otimes {\mathcal F}_2) \leq \varphi(\text{I}(X_1, {\mathcal F}_1), \text{I}(X_2, {\mathcal F}_2))\). Here, \(\varphi\) is some recursion relation, and \({\mathcal F}_1 \otimes {\mathcal F}_2\) is the product of normal bases. As a consequence, the authors obtain that for any finite-dimensional compacta \(X_j\) for \(j = 1, 2\), \({\text{Ind}} (X_1\times X_2) \leq \varphi ({\text{Ind}} X_1, {\text{Ind}} X_2)\).
scientific article; zbMATH DE number 6893992