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The connected Vietoris powerlocale

Vickers, Steven (2009) The connected Vietoris powerlocale. Topology and its Applications, 156. pp. 1886-1910. ISSN 0166-8641

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URL of Published Version: http://www.elsevier.com/locate/topol

Identification Number/DOI: http://dx.doi.org/10.1016/j.topol.2009.03.043

The connected Vietoris powerlocale is defined as a strong monad Vc on the category of locales. VcX is a sublocale of Johnstone's Vietoris powerlocale VX, a localic analogue of the Vietoris hyperspace, and its points correspond to the weakly semifitted sublocales of X that are “strongly connected”. A product map ×:VcX×VcY→Vc(X×Y) shows that the product of two strongly connected sublocales is strongly connected. If X is locally connected then VcX is overt. For the localic completion of a generalized metric space Y, the points of are certain Cauchy filters of formal balls for the finite power set with respect to a Vietoris metric.
Application to the point-free real line gives a choice-free constructive version of the Intermediate Value Theorem and Rolle's Theorem.

The work is topos-valid (assuming natural numbers object). Vc is a geometric construction

Type of Work:Article
Date:2009 (Publication)
School/Faculty:Schools (1998 to 2008) > School of Computer Science
Department:Computer Science
Subjects:Q Science (General)
QA75 Electronic computers. Computer science
QA Mathematics
Institution:University of Birmingham
Copyright Holders:Elsevier
ID Code:182
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