Entropy Dimensions of the Hyperspace of Compact Sets

Let (X; ) be a metric space, let (K(X); e ) denote the space of nonempty compact subsets of X with the Hausdor metric, and let E X. The purpose of this paper is to investigate the relationships between the entropy dimensions of E and of K(E). 1 The de nition and properties of the entropy indices. Given a separable metric space (X; ), let K(X) denote the set of non-empty compact subsets of X . De ne a metric e on K(X) as follows: For A;B 2 K(X) let e (A;B) = maxfsup x2A fdist(x;B)g; sup y2B fdist(y;A)gg: The space (K(X); e ) is called the Hausdor metric space, or hyperspace, associated with X and inherits several nice geometrical properties from X . For example, K(X) is complete whenever X is complete and K(X) is compact whenever X is compact. A discussion of the Hausdor metric including proofs of the above is in [Ed] section 2. 4. To avoid confusion between metric spaces and their corresponding hyperspaces, tildes will be used to denote reference to the hyperspace. So for example, if A X is compact and " > 0, then e B"(A) K(X) denotes the closed ball of radius " about the set A. The upper and lower entropy dimensions, denoted b and b respectively, are de ned in terms of the upper and lower entropy indices denoted and . In this section the upper and lower entropy indices will be de ned in a way suitable for investigating in nite dimensional spaces and some of their basic properties will be developed. The earliest references on the entropy indices are [Bou] and [PonSc]. The indices were generalized and studied on