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Located sets and reverse mathematics

Published online by Cambridge University Press:  12 March 2014

Mariagnese Giusto  and
Stephen G. Simpson
Mariagnese Giusto
Affiliation:
Dip. di Matematica, Universita di Torino, Via Carlo Alberto 10, 10123 Torino, Italy E-mail: giusto@dm.unito.it
Stephen G. Simpson
Affiliation:
Department of Mathematics, Pennsylvania State University, State College. Pennsylvania 16802, USA E-mail: simpson@math.psu.edu
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Abstract

Let X be a compact metric space. A closed set K ⊆ X is located if the distance function d(x, K) exists as a continuous real-valued function on X; weakly located if the predicate d(x, K) > r is allowing parameters. The purpose of this paper is to explore the concepts of located and weakly located subsets of a compact separable metric space in the context of subsystems of second order arithmetic such as RCA0, WKL0 and ACA0. We also give some applications of these concepts by discussing some versions of the Tietze extension theorem. In particular we prove an RCA0 version of this result for weakly located closed sets.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 65 , Issue 3 , September 2000 , pp. 1451 - 1480
Copyright
Copyright © Association for Symbolic Logic 2000

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