Jayne and Rogers proved that every function from an analytic space into a separable metrizable space is decomposable into countably many continuous functions with closed domains if and only if the… (More)

It is known that infinitely many Medvedev degrees exist insid e the Muchnik degree of any nontrivial Π1 subset of Cantor space. We shed light on the fine structures in side these Muchnik degrees… (More)

Consider a randomness notion C. A uniform test in the sense of C is a total computable procedure that each oracle X produces a test relative to X in the sense of C. We say that a binary sequence Y is… (More)

We introduce the point degree spectrum of a represented space as a substructure of the Medvedev degrees, which integrates the notion of Turing degrees, enumeration degrees, continuous degrees, and so… (More)

Every computable function has to be continuous. To develop computability theory of discontin-uous functions, we study low levels of the arithmetical hierarchy of nonuniformly computable functions on… (More)

We study functions from reals to reals which are uniformly degree invariant from Turing equivalence to many-one equivalence, and we compare them “on a cone”. We prove that they are in one-to-one… (More)

The notion of immunity is useful to classify degrees of noncomputability. Meanwhile, the notion of immunity for topological spaces can be thought of as an opposite notion of density. Based on this… (More)

The strong measure zero sets of reals have been widely studied in the context of set theory of the real line. The notion of strong measure zero is straightforwardly effectivized. A set of reals is… (More)

We propose studying uniform Kurtz randomness, which is the uniform relativization of Kurtz randomness. This notion has more natural properties than the usual relativization. For instance, van… (More)