Journal of Symbolic Logic 65 (3):1451-1480 (2000)

Let X be a compact metric space. A closed set K $\subseteq$ X is located if the distance function d exists as a continuous real-valued function on X; weakly located if the predicate d $>$ r is $\Sigma^0_1$ 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 RCA$_0$, WKL$_0$ and ACA$_0$. We also give some applications of these concepts by discussing some versions of the Tietze extension theorem. In particular we prove an RCA$_0$ version of this result for weakly located closed sets.
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Mass Problems and Randomness.Stephen G. Simpson - 2005 - Bulletin of Symbolic Logic 11 (1):1-27.
The Polarized Ramsey’s Theorem.Damir D. Dzhafarov & Jeffry L. Hirst - 2009 - Archive for Mathematical Logic 48 (2):141-157.
Separation and Weak König's Lemma.A. James Humphreys & Stephen G. Simpson - 1999 - Journal of Symbolic Logic 64 (1):268-278.

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