This paper continues the study of the metric topology on that was introduced by S. Binns. This topology is induced by a directional metric where the distance from to is given by
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 and that under it the functions and are continuous.
We also investigate the scalar multiplication operation that was introduced by Binns. The multiplication of a real by an element represents a dilution of the information in by a factor of .
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.
"Compressibility and Kolmogorov Complexity." Notre Dame J. Formal Logic 54 (1) 105 - 123, 2013. https://doi.org/10.1215/00294527-1731416