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Hausdorff measures and two point set extensions. (English) Zbl 0910.28005

A planar set \(A\) is called: \(\bullet\) a two point set if every line intersects \(A\) in exactly two points; \(\bullet\) a partial two point set if every line intersects \(A\) in at most two points; \(\bullet\) extendable if \(A\) is a subset of some two point set. The paper contains some interesting results concerning extendable sets. In the first part the authors prove that there exist small nonextendable compact subsets of the unit circle \(S^1\), i.e., of linear Lebesgue measure zero (logarithmic capacity zero or Hausdorff dimension zero, respectively). In the second part the authors prove that under Martin’s Axiom any \(\sigma\)-compact partial two point set such that its square has Hausdorff \(1\)-measure zero is extendable. Moreover, it is shown extendable sets form a dense \(G_{\delta}\) subset in the space \({\mathcal K}(S^1)\) of nonempty compact subsets of the \(S^1\).

MSC:

28A78 Hausdorff and packing measures
28A05 Classes of sets (Borel fields, \(\sigma\)-rings, etc.), measurable sets, Suslin sets, analytic sets
54H05 Descriptive set theory (topological aspects of Borel, analytic, projective, etc. sets)
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