We extend Kawamura and Cook's framework for computational complexity for operators in analysis. This model is based on second-order complexity theory for functionals on the Baire space, which is lifted to metric spaces via representations. Time is measured in the length of the input encodings and the output precision. We propose the notions of complete and regular representations. Completeness is proven to ensure that any computable function has a time bound. Regularity relaxes Kawamura and Cook's notion of a second-order representation, while still guaranteeing fast computability of the length of encodings. We apply these notions to investigate relationships between metric properties of a space and existence of representations that render the metric bounded-time computable. We show that time bounds for the metric can straightforwardly be translated into size bounds of compact subsets of the space. Conversely, for compact spaces and for Banach spaces we construct admissible complete regular representations admitting fast computation of the metric and short encodings. Here it is necessary to trade time bounds off against length of encodings.

We define and study hierarchies of topological spaces induced by the classical Borel and Luzin hierarchies of sets. Our hierarchies are divided into two classes: hierarchies of countably based spaces induced by their embeddings into Pω, and hierarchies of spaces (not necessarily countably based) induced by their admissible representations. We concentrate on the non-collapse property of the hierarchies and on the relationships between hierarchies in the two classes.

The compact-open topology on the set of continuous functionals from the Baire space to the natural numbers is well known to be zero-dimensional. We prove that the closely related sequential topology on this set is not even regular. The sequential topology arises naturally as the topology carried by the exponential formed in various cartesian closed categories of topological spaces. Moreover, we give an example of an effectively open subset of that violates regularity. The topological properties of are known to be closely related to an open problem in Computable Analysis. We also show that the sequential topology on the space of continuous real-valued functions on a Polish space need not be regular.

We prove two results for the sequential topology on countable products of sequential topological spaces. First we show that a countable product of topological quotients yields a quotient map between the product spaces. Then we show that the reflection from sequential spaces to its subcategory of monotone ω-convergence spaces preserves countable products. These results are motivated by applications to the modelling of computation on non-discrete spaces.

We propose compactly generated monotone convergence spaces as a well-behaved topological generalisation of directed-complete partial orders (dcpos). The category of such spaces enjoys the usual properties of categories of ‘predomains’ in denotational semantics. Moreover, such properties are retained if one restricts to spaces with a countable pseudobase in the sense of E. Michael, a fact that permits connections to be made with computability theory, realizability semantics and recent work on the closure properties of topological quotients of countably based spaces (qcb spaces). We compare the standard domain-theoretic constructions of products and function spaces on dcpos with their compactly generated counterparts, showing that these agree in important cases, though not in general.