# The Theorem of Pontrjagin-Schnirelmann and Approximate Sequences

### From NZJM

**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

**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 which represent compact metric spaces *X* as approximate resolutions : *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