On the completeness of a certain metric space with an application to Blaschke’s selection theorem

Abstract

2. Preliminaries. Let K be a metric space with elements x, y, • • • and distance function d(x, y). A sequence xi, x2, • • • in K such that ^2T^(Xiy Xi+i) converges has been called an absolutely convergent sequence by MacNeille [7, p. 192]. Every absolutely convergent sequence is a Cauchy sequence, and every Cauchy sequence contains absolutely convergent subsequences. Let i£* be a metric space whose elements X, F, • • • are the closed, bounded, and non-null subsets of K, and whose distance function D(X, Y) is the Hausdorff distance between the sets X and Y (see Hausdorff [5, pp. 145-146] and Kuratowski [6, pp. 89-90]).