The Theorem of Pontrjagin-Schnirelmann and Approximate Sequences

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New Zealand Journal of Mathematics

Vol. 38, (2008), Pages 121-128

I. Ibedou

Department of Mathematics

Faculty of Science

Benha University

Benha

EGYPT

mailto:ismail.ibedou@yahoo.com

T. Miyata

Department of Mathematics and Informatics

Graduate School of Human Development and Environment

Kobe University

Kobe, 657-8501

JAPAN

mailto:tmiyata@kobe-u.ac.jp



Abstract In this paper we obtain a modified version of the classical theorem of Pontrjagin and Schnirelmann concerning the covering dimension of a compact metric space and the box-counting dimensions associated with the metrics on the space. T. Miyata and T. Watanabe defined box-counting dimension for approximate sequences \mathbf X which represent compact metric spaces X as approximate resolutions \mathbf p: X \rightarrow \mathbf X. We show that the covering dimension equals the infimum of the box-counting dimensions associated with the approximate resolutions of the space.

Keywords Covering dimension, approximate sequence, box-counting dimension

Classification (MSC2000) Primary: 54F45; Secondary: 54C56, 28A80

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