Bogatyi, S. A.; Karpov, A. N. Bruijning-Nagata and Hashimoto-Hattori characteristics of covering dimension revisited. (English) Zbl 1123.54012 Math. Notes 79, No. 3, 327-334 (2006); translation from Mat. Zametki 79, No. 3, 353-361 (2006). The functions \(\Delta_k(X)\) and \(\ast_k(X)\) for a topological space \(X\) and \(k\in{\mathbb N}\) were introduced by Bruijning and Nagata, respectively. They proved that for every infinite normal space \(X\) with \(\dim X=n\) and every \(k\in{\mathbb N}\), \(\Delta_k(X)=2^k-1\) if \(k\leq n+1\); \(\Delta_k(X)=\sum_{m=1}^{n+1}C^m_k\) if \(k\geq n+1\). By use of this equation, they gave a characterization of the covering dimension for an infinite normal space \(X\) such as \(\dim X=\lim_{k\to\infty}\frac{\log\Delta_k(X)}{\log k}-1\). Furthermore, Hashimoto and Hattori obtained similar equations for \(\ast_k(X)\): if \(X\) is an infinite normal space with \(\dim X=n\), then \(\ast_k(X)=k\cdot2^{k-1}\) if \(k\leq n+1\); \(\ast_k(X)=\sum_{m=1}^{n+1}mC^m_k\) if \(k\geq n+1\), and for every infinite normal space \(X\) it holds that \(\dim X=\lim_{k\to\infty}\frac{\log\ast_k(X)}{\log k}-1\). In this paper, the authors define new dimension functions \(\Delta_k^l(X)\) and \(\ast_k^l(X)\) for a topological space \(X\) and \(k,l\in{\mathbb N}\). As main results, they obtain the following equations including the above formulas: if \(X\) is an infinite normal space with \(\dim X=n\), then \(\Delta_k^l(X)=(l+1)^k-l^k\) if \(k\leq n+1\); \(\Delta_k^l(X)=\sum_{m=1}^{n+1}(l^m-(l-1)^m)C^m_k\) if \(k\geq n+1\) and \(\ast_k^l(X)=k\cdot(l+1)^{k-1}\) if \(k\leq n+1\); \(\ast_k^l(X)=\sum_{m+1}^{n+1}ml^{m-1}C^m_k\) if \(k\geq n+1\), and for every infinite normal space \(X\) and \(l\in{\mathbb N}\) \(\dim X=\lim_{k\to\infty}\frac{\log\Delta^l_k(X)}{\log k}-1 =\lim_{k\to\infty}\frac{\log\ast^l_k(X)}{\log k}-1\). Reviewer: Kohzo Yamada (Shizuoka) Cited in 1 Document MSC: 54F45 Dimension theory in general topology Keywords:covering dimension; dimension function; Bruijning-Nagata function; Nagata’s star-index; regular refinement of a cover; weak star refinement PDFBibTeX XMLCite \textit{S. A. Bogatyi} and \textit{A. N. Karpov}, Math. Notes 79, No. 3, 327--334 (2006; Zbl 1123.54012); translation from Mat. Zametki 79, No. 3, 353--361 (2006) Full Text: DOI References: [1] J. Bruijning and J. Nagata, ”A characterization of covering dimension by use of {\(\Delta\)} k(X),” Pacific J. Math., 80 (1979), no. 1, 1–8. · Zbl 0407.54029 [2] J. Nagata, ”Some questions II,” Questions Answers Gen. Topology, 2 (1984), 2–13. · Zbl 0586.54036 [3] K. Hashimoto and Y. Hattori, ”On Nagata’s star-index *k(X),” Topology Appl., 122 (2002), 201–204. · Zbl 0992.54031 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.