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.
@article{bwmeta1.element.bwnjournal-article-fmv157i1p43bwm,
author = {Jan Dijkstra and Kenneth Kunen and Jan van Mill},
title = {Hausdorff measures and two point set extensions},
journal = {Fundamenta Mathematicae},
volume = {158},
year = {1998},
pages = {43-60},
zbl = {0910.28005},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-fmv157i1p43bwm}
}
Dijkstra, Jan; Kunen, Kenneth; van Mill, Jan. Hausdorff measures and two point set extensions. Fundamenta Mathematicae, Tome 158 (1998) pp. 43-60. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-fmv157i1p43bwm/
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