Liminf convergence in \Omega\Gamma33432/-630
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Abstract:
The aim of this article is twofold. From a mathematical perspective we present a notion of convergence which is suitably general such as to include the convergence of chains to their least upper bounds in preordered sets, and the convergence of Cauchy sequences to their metric limits in metric spaces. Rather than presenting this theory from a purely mathematical perspective however, we will use it to introduce a simple-minded domain theory based on a generic notion of approximation. We might hope that this is not the only use of the concepts we present, although it is the one that motivated us in the first place. One possible kind of approximation one uses in domain theory as it is used in the study of denotational semantics of programming languages is the binary one that we have in preorders: either an element is below another in the preordering, or it is not. Another kind of approximation is the metric one, where we do not just say whether one element approximates another, but to which degree it does so, with a non-negative real number. It turns out that we can separate out from a large part of domain theory considerations about a particular notion of approximation, and just state a few axioms that should hold about a notion of approximation. In this general theory, which encompasses preorders and metric spaces among many other kinds of structures, we can then do general domain theory. The requirements for our notion of approximation turns out to be
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