A topological characterization of random sequences

Abstract

The set of random sequences is large in the sense of measure, but small in the sense of category. This is the case when we regard the set of infinite sequences over a finite alphabet as a subset of the usual Cantor space. In this note we will show that the above result depends on the topology chosen. To this end we will use a relativization of the Cantor topology, the U δ -topology introduced by Staiger [RAIRO Inform. Théor. 21 (1987) 147–173]. This topology is also metric, but the distance between two sequences does not depend on their longest common prefix (Cantor metric), but on the number of their common prefixes in a given language U. The resulting space is complete, but not always compact. We will show how to derive a computable set U from a universal Martin-Löf test such that the set of non-random sequences is nowhere dense in the U δ -topology. As a byproduct we obtain a topological characterization of the set of random sequences. We also show that the Law of Large Numbers, which fails with respect to the usual topology, is true for the U δ -topology.