# The Theorem of Pontrjagin-Schnirelmann and Approximate Sequences

### New Zealand Journal of Mathematics

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

I. Ibedou

Department of Mathematics

Faculty of Science

Benha University

Benha

EGYPT

T. Miyata

Department of Mathematics and Informatics

Graduate School of Human Development and Environment

Kobe University

Kobe, 657-8501

JAPAN

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