Let X be a compact metric space. A closed set K 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 # 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.

investigating the strength of set existence axioms needed for separable Banach space theory. We show that the separation theorem foropenconvexsetsisequivalenttoWKL0 over RCA0. Weshow that the separation theorem for separably closed convex sets is equivalent to ACA0 over RCA0. Our strategy for proving these geometrical Hahn–Banach theorems is to reduce to the finite-dimensional case by means of a compactness argument. 1.