Computable Bases

Abstract

In computable analysis typically topological spaces with countable bases are considered. The Theorem of Kreitz-Weihrauch implies that the subbase representation of a second-countable $T_0$ space is admissible with respect to the topology that the subbase generates. We consider generalizations of this setting to bases that are representable, but not necessarily countable. We introduce the notions of a computable presubbase and a computable prebase. We prove a generalization of the Theorem of Kreitz-Weihrauch for the presubbase representation that shows that any such representation is admissible with respect to the topology generated by compact intersections of the presubbase elements. For computable prebases we obtain representations that are admissible with respect to the topology that they generate. These concepts provide a natural way to investigate many topological spaces that have been studied in computable analysis. The benefit of this approach is that topologies can be described by their usual subbases and standard constructions for such subbases can be applied. Finally we discuss a Galois connection between presubbases and representations of $T_0$ spaces that indicates that presubbases and representations offer particular views on the same mathematical structure from different perspectives.