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Relative formal topology: the binary positivity predicate comes first

Published online by Cambridge University Press:  16 November 2011

SILVIO VALENTINI
SILVIO VALENTINI*
Affiliation:
Dipartimento di Matematica Pura ed Applicata, Università di Padova, via Trieste n. 63, I–35121 Padova, Italy Email: silvio@math.unipd.it
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Abstract

In this paper we introduce relative formal topology so that we can deal constructively with the closed subsets of a topological space as well as with the open ones. In fact, within standard formal topology, the cover relation allows the definition of the formal opens, which are supposed to act as the formal counterparts of the open subsets, and within balanced formal topology, the binary positivity predicate allows the definition of the formal closed subsets, which are supposed to act as the formal counterparts of the closed subsets. However, these approaches are only fully satisfactory (according to the criterion introduced by the author in Valentini (2005)) if we can provide an adequate formalisation of the cover relation and the positivity predicate. But current formalisations fail in this respect since some intuitionistically valid relations between the open and closed subsets of a concrete topological space cannot be expressed. Our central result is that we can solve this problem through a generalisation of the standard cover relation together with a binary positivity predicate satisfying the positivity axiom.

Type
Paper
Information
Mathematical Structures in Computer Science , Volume 22 , Issue 1 , February 2012 , pp. 69 - 102
Copyright
Copyright © Cambridge University Press 2011

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