Admissible domain representations of convergence spaces

In this paper we consider admissible domain representations of convergence spaces. A convergence space is a pair (X, →X) where X is a set and →X is a binary relation between nets on X and elements of X. We study in particular the case when (X, →X) is a weak κ-convergence space, which roughly means that →X is a relation satisfying a generalisation of the Kuratowski limit axioms to cardinality κ. As a framework for our study of weak κ-convergence spaces and the related class of weak convergence spaces we use admissible domain representations. A domain representation D of a space X is λ-admissible if, in principle, all other λ-based domain representations E of X can be reduced to D via a continuous function from E to D. We present two major results. First we show that the category of weak κconvergence spaces is cartesian closed. The second result is that the category of weak κ-convergence spaces that have a dense, λ-admissible, κ-continuous and α-based consistently complete domain representation is cartesian closed when α ≤ λ ≥ κ, thus generalising the results of [Sch01]. As natural corollaries we obtain corresponding results for weak convergence spaces. 1