Characterizations of topological dimension by use of normal sequences of finite open covers and PontrjaginSchnirelmann theorem

 Kato Hisao
 Institute of Mathematics, University of Tsukuba

 Matsumoto Masahiro
 Institute of Mathematics, University of Tsukuba
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Description
In 1932, Pontrjagin and Schnirelmann [15] proved the classical theorem which characterizes topological dimension by use of boxcounting dimensions. They proved their theorem by use of geometric arguments in some Euclidean spaces. In this paper, by use of dimensional theoretical techniques in an abstract topological space, we investigate strong relations between metrics of spaces and boxcounting dimensions. First, by use of the numerical information of normal sequences of finite open covers of a space X, we prove directly the following theorem characterizing topological dimension dim X. <br/> Theorem 0.1. Let X be a nonempty separable metric space. Then <br/> dim X = min { $¥liminf_{i¥to ¥infty} ¥frac{¥log_{3}{¥mathscr U}_{i}}{i}$  {$¥mathscr U$_{i}}_{i=1}^{∞} is a normal starsequence of finite open covers of X and a development of X} <br/> = min { $¥liminf_{i¥to ¥infty} ¥frac{¥log_{2}{¥mathscr U}_{i}}{i}$  {$¥mathscr U$_{i}}_{i=1}^{∞} is a normal deltasequence of finite open covers of X and a development of X }.<br/> Next, we study boxcounting dimensions dim_{B}(X,d) by use of AlexandroffUrysohn metrics d induced by normal sequences. We show that the above theorem implies PontrjaginSchnirelmann theorem. The proof is different from the one of Pontrjagin and Schnirelmann (see [15]). By use of normal sequences, we can construct freely metrics d which control the values of log N(ε,d)/log ε. In particular, we can construct chaotic metrics with respect to the determination of the boxcounting dimensions as follows. <br/> Theorem 0.2. Let X be an infinite separable metric space. For any ∞ ≥ α ≥ dim X, there is a totally bounded metric d_{α} on X such that <br/> [α,∞] = { $¥liminf_{k¥to ¥infty}¥frac{¥log N(¥epsilon_{k},d_{¥alpha})}{¥log ¥epsilon_{k}}$  {ε_{k}}_{k=1}^{∞} is a decreasing sequence of positive numbers with lim_{k→∞} ε_{k}=0},<br/> where N(ε_{k}, d_{α}) = min{$¥mathscr U$ $¥mathscr U$ is a finite open cover of X with mesh_{d<sub>α}</sub>($¥mathscr U$) ≤ ε_{k}}. In particular, dim_{B}(X,d_{α}) = α.
Journal

 Journal of the Mathematical Society of Japan

Journal of the Mathematical Society of Japan 63 (3), 919976, 2011
The Mathematical Society of Japan
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Details 詳細情報について

 CRID
 1390282680092645888

 NII Article ID
 10029321958

 NII Book ID
 AA0070177X

 ISSN
 18811167
 18812333
 00255645

 NDL BIB ID
 11170261

 Text Lang
 en

 Data Source

 JaLC
 NDL
 Crossref
 CiNii Articles
 KAKEN

 Abstract License Flag
 Disallowed