Computably Based Locally Compact Spaces
<|reference_start|>Computably Based Locally Compact Spaces: ASD (Abstract Stone Duality) is a re-axiomatisation of general topology in which the topology on a space is treated, not as an infinitary lattice, but as an exponential object of the same category as the original space, with an associated lambda-calculus. In this paper, this is shown to be equivalent to a notion of computable basis for locally compact sober spaces or locales, involving a family of open subspaces and accompanying family of compact ones. This generalises Smyth's effectively given domains and Jung's strong proximity lattices. Part of the data for a basis is the inclusion relation of compact subspaces within open ones, which is formulated in locale theory as the way-below relation on a continuous lattice. The finitary properties of this relation are characterised here, including the Wilker condition for the cover of a compact space by two open ones. The real line is used as a running example, being closely related to Scott's domain of intervals. ASD does not use the category of sets, but the full subcategory of overt discrete objects plays this role; it is an arithmetic universe (pretopos with lists). In particular, we use this subcategory to translate computable bases for classical spaces into objects in the ASD calculus.<|reference_end|>
@article{taylor2005computably,
    title={Computably Based Locally Compact Spaces},
    author={Paul Taylor},
    journal={Logical Methods in Computer Science, Volume 2, Issue 1 (March 7,
  2006) lmcs:2255},
    year={2005},
    doi={10.2168/LMCS-2(1:1)2006},
    archivePrefix={arXiv},
    eprint={math/0512110},
    primaryClass={math.GN cs.LO math.CT}
}