Infinite limits and R-recursive functions

In this paper we use infinite limits to define R-recursive functions. We prove that the class of R-recursive functions is closed under this operation

The descriptive theory of represented spaces

This is a survey on the ongoing development of a descriptive theory of represented spaces, which is intended as an extension of both classical and effective descriptive set theory to deal with both sets and functions between represented spaces. Most material is from work-in-progress, and thus there may be a stronger focus on projects involving the author than an objective survey would merit.Comment: survey of work-in-progres

Functional first order definability of LRTp

The language LRTp is a non-deterministic language for exact real number computation. It has been shown that all computable rst order relations in the sense of Brattka are denable in the language. If we restrict the language to single-valued total relations (e.g. functions), all polynomials are denable in the language. This paper is an expanded version of [12] in which we show that the non-deterministic version of the limit operator, which allows to dene all computable rst order relations, when restricted to single-valued total inputs, produces single-valued total outputs. This implies that not only the polynomials are denable in the language but also allcomputable rst order functions

On the topological aspects of the theory of represented spaces

Represented spaces form the general setting for the study of computability derived from Turing machines. As such, they are the basic entities for endeavors such as computable analysis or computable measure theory. The theory of represented spaces is well-known to exhibit a strong topological flavour. We present an abstract and very succinct introduction to the field; drawing heavily on prior work by Escard\'o, Schr\"oder, and others. Central aspects of the theory are function spaces and various spaces of subsets derived from other represented spaces, and -- closely linked to these -- properties of represented spaces such as compactness, overtness and separation principles. Both the derived spaces and the properties are introduced by demanding the computability of certain mappings, and it is demonstrated that typically various interesting mappings induce the same property.Comment: Earlier versions were titled "Compactness and separation for represented spaces" and "A new introduction to the theory of represented spaces