Located Sets and Reverse Mathematics

LetX be a compact metric space. A closed setK ⊆ X is located if the distance function d(x,K) exists as a continuous realvalued function on X ; weakly located if the predicate d(x,K) > r is Σ1 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.