by **
C. Richter**

**Preprint series:**
99-44, Reports on Analysis

**MSC:**- 54E45 Compact (locally compact) metric spaces
- 52C99 None of the above but in this section

**Abstract:** Given a compact metric space $(X,d)$ so-called controllable coverings

of $X$ are defined by a uniformity condition in terms of entropy numbers.

This geometric condition is closely related to Kolmogoroff's entropy

function of $X$.

We discuss the following question motivated by a problem from approximation

theory: Let $d^\prime$ be another metric on $X$ such that $(X,d^\prime)$ is

compact. When do $d$ and $d^\prime$ admit the same controllable coverings?

We present sufficient as well as necessary conditions by the aid of

Kolmogoroff's entropy function and, in the end, give a construction for a

new metric $d^\prime$ essentially different from the original $d$ which

gives rise to the same controllable coverings.

**Keywords:** *copact metric space, covering, entropy number, Kolmogoroff's entropy function*

**Upload:** 1999-07-27

**Update:** 1999-07-27

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