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Steven Vickers [8]Steven J. Vickers [1]
  1.  22
    Topology via Logic.P. T. Johnstone & Steven Vickers - 1991 - Journal of Symbolic Logic 56 (3):1101.
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  2.  12
    Positivity relations on a locale.Francesco Ciraulo & Steven Vickers - 2016 - Annals of Pure and Applied Logic 167 (9):806-819.
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  3.  37
    Compactness in locales and in formal topology.Steven Vickers - 2006 - Annals of Pure and Applied Logic 137 (1-3):413-438.
    If a locale is presented by a “flat site”, it is shown how its frame can be presented by generators and relations as a dcpo. A necessary and sufficient condition is derived for compactness of the locale . Although its derivation uses impredicative constructions, it is also shown predicatively using the inductive generation of formal topologies. A predicative proof of the binary Tychonoff theorem is given, including a characterization of the finite covers of the product by basic opens. The discussion (...)
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  4.  22
    Sublocales in Formal Topology.Steven Vickers - 2007 - Journal of Symbolic Logic 72 (2):463 - 482.
    The paper studies how the localic notion of sublocale transfers to formal topology. For any formal topology (not necessarily with positivity predicate) we define a sublocale to be a cover relation that includes that of the formal topology. The family of sublocales has set-indexed joins. For each set of base elements there are corresponding open and closed sublocales, boolean complements of each other. They generate a boolean algebra amongst the sublocales. In the case of an inductively generated formal topology, the (...)
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  5.  30
    Partial Horn logic and cartesian categories.Erik Palmgren & Steven J. Vickers - 2007 - Annals of Pure and Applied Logic 145 (3):314-353.
  6.  15
    Continuity and geometric logic.Steven Vickers - 2014 - Journal of Applied Logic 12 (1):14-27.
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  7.  43
    Cosheaves and connectedness in formal topology.Steven Vickers - 2012 - Annals of Pure and Applied Logic 163 (2):157-174.
  8.  29
    A localic theory of lower and upper integrals.Steven Vickers - 2008 - Mathematical Logic Quarterly 54 (1):109-123.
    An account of lower and upper integration is given. It is constructive in the sense of geometric logic. If the integrand takes its values in the non-negative lower reals, then its lower integral with respect to a valuation is a lower real. If the integrand takes its values in the non-negative upper reals, then its upper integral with respect to a covaluation and with domain of integration bounded by a compact subspace is an upper real. Spaces of valuations and of (...)
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  9.  10
    The localic compact interval is an Escardó‐Simpson interval object.Steven Vickers - 2017 - Mathematical Logic Quarterly 63 (6):614-629.
    The locale corresponding to the real interval [ − 1, 1] is an interval object, in the sense of Escardó and Simpson, in the category of locales. The map, mapping a stream s of signs ±1 to, is a proper localic surjection; it is also expressed as a coequalizer. The proofs are valid in any elementary topos with natural numbers object.
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