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From Sets and Types to Topology and AnalysisTowards practicable foundations for constructive mathematics$
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Laura Crosilla and Peter Schuster

Print publication date: 2005

Print ISBN-13: 9780198566519

Published to Oxford Scholarship Online: September 2007

DOI: 10.1093/acprof:oso/9780198566519.001.0001

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SOME CONSTRUCTIVE ROADS TO TYCHONOFF

SOME CONSTRUCTIVE ROADS TO TYCHONOFF

Chapter:
(p.223) 14 SOME CONSTRUCTIVE ROADS TO TYCHONOFF
Source:
From Sets and Types to Topology and Analysis
Author(s):

Steven Vickers

Publisher:
Oxford University Press
DOI:10.1093/acprof:oso/9780198566519.003.0014

This chapter discusses the Tychonoff Theorem with respect to point-free topology from the point of view of both topos-valid and predicative mathematics. A new proof of the infinitary Tychonoff Theorem is given using predicative, choice-free methods for possibly undecidable index set. It yields a complete description of the finite basic covers of the product. In contrast to the formal-topological treatment previously given by Negri and Valentini, who followed Coquand's first paper on this subject, the index set of the product under consideration need not be decidable in this chapter's proof. While a more recent approach by Coquand is based on the assumption that each locale under consideration is presented by a distributive lattice of generators, this chapter does not assume the presence of any such presentation. In passing, it highlights the differences and connections between the point-set and the point-free approaches to topology, and between the major varieties of the latter, locale theory, and formal topology.

Keywords:   point-free topology, topos theory, predicative mathematics, choice-free mathematics

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