Taylor, Paul - Computably Based Locally Compact Spaces

lmcs:2255 - Logical Methods in Computer Science, March 7, 2006, Volume 2, Issue 1 - https://doi.org/10.2168/LMCS-2(1:1)2006
Computably Based Locally Compact Spaces

Authors: Taylor, Paul

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.

Volume: Volume 2, Issue 1
Published on: March 7, 2006
Submitted on: September 3, 2004
Keywords: Mathematics - General Topology,Computer Science - Logic in Computer Science,Mathematics - Category Theory,54D45, 03D45 (Primary), 06B35, 54D30, 68N18 (Secondary)


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