Hausdorff measures and two point set extensions
Volume 157 / 1998
Fundamenta Mathematicae 157 (1998), 43-60 DOI: 10.4064/fm-157-1-43-60
We investigate the following question: under which conditions is a σ-compact partial two point set contained in a two point set? We show that no reasonable measure or capacity (when applied to the set itself) can provide a sufficient condition for a compact partial two point set to be extendable to a two point set. On the other hand, we prove that under Martin's Axiom any σ-compact partial two point set such that its square has Hausdorff 1-measure zero is extendable.