{
  "id": "math/0512110",
  "version": "v4",
  "published": "2005-12-05T22:31:32.000Z",
  "updated": "2006-03-07T11:16:13.000Z",
  "title": "Computably Based Locally Compact Spaces",
  "authors": [
    "Paul Taylor"
  ],
  "comment": "70pp, LaTeX2e, uses diagrams.sty; Accepted for \"Logical Methods in Computer Science\" LMCS-2004-19; see http://www.cs.man.ac.uk/~pt/ASD for related papers. ACM-class: F.4.1",
  "journal": "LMCS 1 (4:1) 2006",
  "doi": "10.2168/LMCS-1(4:1)2006",
  "categories": [
    "math.GN",
    "cs.LO",
    "math.CT"
  ],
  "abstract": "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.",
  "revisions": [
    {
      "version": "v4",
      "updated": "2006-03-07T11:16:13.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "54D45",
      "03D45",
      "06B35",
      "54D30",
      "68N18"
    ],
    "keywords": [
      "locally compact spaces",
      "jungs strong proximity lattices",
      "overt discrete objects plays",
      "locally compact sober spaces",
      "abstract stone duality"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "LaTeX",
      "pages": 70,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}