Compressibility and Kolmogorov Complexity

This paper continues the study of the metric topology on 2(N) that was introduced by S. Binns. This topology is induced by a directional metric where the distance from Y is an element of 2(N) to X is an element of 2(N) is given by lim sup(n) C(X (sic) n (sic) Y (sic) n)/n This definition is closely related to the notions of effective Hausdorff and packing dimensions. Here we establish that this is a path-connected topology on 2(N) and that under it the functions X bar right arrow dim(H) X and X bar right arrow dim(p) X are continuous. We also investigate the scalar multiplication operation that was introduced by Binns. The multiplication of a real X is an element of 2N by an element alpha is an element of [0, 1] represents a dilution of the information in X by a factor of alpha. Our main result is to show that every regular real is the dilution of a real of Hausdorff dimension 1. That is, that the information in every regular real can be maximally compressed.