Abstract

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.