Note on the topological structure of random strings
The natural way to define an effective topology on the set of finite strings is to start with a recursive order \(\leq\) and to define for each string \(v\) its basic open set as \(U_ v=\{w | v \leq w\}\). The topological size of random Kolmogorov strings is investigated with respect to a lot of topologies based, as above, on natural orderings of finite strings. The results are of the following types (1) the set of nonrandom strings is dense and (2) the set of random strings is recursively rare with the notably exception of the topology derived from the infix ordering. In this case, for strings over an alphabet with at least three symbols, the random strings form a dense set, while for binary strings the question remains open.
scientific article; zbMATH DE number 178044
scientific article