Recursion and topology on 2<=omega for possibly infinite computations

In the context of possibly infinite computations yielding finite or infinite (binary) outputs, the space 2 ≤ω = 2* U 2 ω appears to be one of the most fundamental spaces in Computer Science. Though underconsidered, next to 2 ω , this space can be viewed (Section 3.5.2) as the simplest compact space native to computer science. In this paper we study some of its properties involving topology and computability. Though 2 ≤ω can be considered as a computable metric space in the sense of computable analysis, a direct and self-contained study, based on its peculiar properties related to words, is much illuminating. It is well known that computability for maps 2 ω → 2 ω reduces to continuity with recursive modulus of continuity. With 2 ≤ω , things get less simple. Maps 2 ω → 2 ≤ω or 2 ≤ω → 2 ≤ω induced by input/output behaviours of Turing machines on finite or infinite words-which we call semicomputable maps-are not necessarily continuous but merely lower semicontinuous with respect to the prefix partial ordering on 2 ≤ω . Continuity asks for a stronger notion of computability. We prove for (semi)continuous and (semi)computable maps F: F → O with F, O ∈ {2 ω , 2 ≤ω } a detailed representation theorem (Theorem 81) via functions f:2* → 2* following two approaches: bottom-up from f to F and top-down from F to f.