To Jessica

Blanck, J., 1997: Computability on topological spaces by eeective domain representations. Our aim in this thesis is to study a uniform method to introduce computability on large, usually uncountable, mathematical structures. The method we choose is domain representations using Scott{Ershov domains. Domain theory is a theory of approximations and incorporates a natural computability theory. This provides us with a uniform way to introduce computability on structures that have computable domain representations, by computations on the approximations of the structure. It is shown that large classes of topological spaces have satisfactory domain representations. In particular, all metric spaces are domain representable. It is also shown that the space of compact subsets of a complete metric space can be given a domain representation uniformly from a domain representation of the metric space. Several other classes of topological spaces are shown to have domain representations, although not all of them are suitable for introducing computability. Domain representations of topological spaces are also used to give new proofs of some known results using theorems from domain theory. An example is that we use the xed point theorem for domains to show both Banach's xed point theorem for contractions on complete metric spaces and that every hyperbolic iterated function system has a compact attractor. That Banach's theorem follows from the xed point theorem for domains could be seen as a partial explanation to the fact that both domains and metric spaces have successfully been used to give denotational semantics. Another example is the use of Berger's domain theoretic generalisation of the Kreisel{Lacombe{Schoenneld theorem to show Ceitin's theorem. Acknowledgements I am deeply indebted to my advisor Viggo Stoltenberg-Hansen, who apart from spending endless hours discussing my research and sharing his knowledge also has given me the support of a friend. I would like to thank the members of the group of Mathematical logic at our department and the participants of the Stockholm{ Uppsala logic seminar for establishing a fruitful research climate. I am grateful to John N. Crossley who made it possible for me to spend a couple of months at Monash University, Australia. I thank Ulrich Berger and John V. Tucker for their comments on my rst published paper. I also thank my colleagues at the department and especially my fellow graduate students. Family and friends have also contributed in various ways to make this possible. I would also like to express my deepest gratitude …