Recursive quasi-metric spaces

In computable analysis recursive metric spaces play an important role, since these are, roughly speaking, spaces with computable metric and limit operation. Unfortunately, the concept of a metric space is not powerful enough to capture all interesting phenomena which occur in computable analysis. Some computable objects are naturally considered as elements of asymmetric spaces which are not metrizable. Nevertheless, most of these spaces are T0-spaces with countable bases and thus at least quasi-metrizable. We introduce a definition of recursive quasi-metric spaces in analogy to recursive metric spaces. We show that this concept leads to similar results as in the metric case and we prove that the most important spaces of computable analysis can be naturally considered as recursive quasi-metric spaces. Especially, we discuss some hyper and function spaces.

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