Abstract
We give a summary of the TTE-approach to computable anlysis, as background for a discussion about Borel complexity on represented spaces. We study the hyperspace A(X) of closed subsets of a separable metric space X, and consider the representations representationsof this space, corresponding to the upper Fell topology, lower Fell topology and Fell topology, respectively. All of these representations are Borel equivalent, and admits Borel measurable liftings of the Cantor derivative, if X is compact. However, if X is an uncountable Polish space, the map sending a closed subset to its perfect component, which corresponds to the transfi nitely iterated Cantor derivative, does not have a Borel measurable lifting relative to any of these representations. Finally, we study a representation of the Borel algebra B(X) on a topological space X, reflecting the
way the Borel sets are generated from the open sets. We show that complementation, binary union and countable union all have computable liftings relative to this representation , and we find conditions ensuring that the dual of a continuous function has a continuous lifting. Background from descriptive set theory is provided in an appendix.