Theorie der Darstellungen und ihre Anwendungen
in der konstruktiven Analysis. 

Christoph Kreitz.  
PhD Thesis, Fernuniversität Hagen, 1984. (German) 

Abstract 

We present an approach for a unified theory of effectivity on nondenumerable
sets. It is based on a standard Type 2 theory of computability and
continuity on F={f:N>N} and a theory of representations,
i.e. mappings from F onto some set As an application show how to use these concepts as a foundation of a new kind of constructive analysis. We compare different representations of the real numbers and of the open subsets of the real numbers and investigate which of them are suitable for studying computability in analysis. In particular we consider constructive compactness and investigate the constructivity properties of the standard theorems about continuous functions on compact sets. 

The thesis is written in German. Its contents are essentially covered by the
following publications
 A unified approach to constructive and recursive analysis. Logic Colloquium Aachen, 1984.  Theory of representations. Theoretical Computer Science, 1985.  Representations of the real numbers and the open subsets of the set of real numbers. Annals of pure and applied logic, 1987.  Compactness in constructive analysis revisited. Annals of pure and applied logic, 1987. 
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Bibtex Entry 

@PhDThesis{phd:Kreitz84a, author = "Christoph Kreitz", title = "{Theorie der Darstellungen und ihre Anwendungen in der konstruktiven Analysis}", school = "FernUniversit{\"a}t Hagen, FB Mathematik und Informatik", year = "1984" } 