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Generic Partial Two-Point Sets Are Extendable

Published online by Cambridge University Press:  20 November 2018

Jan J. Dijkstra
Jan J. Dijkstra*
Affiliation:
Department of Mathematics The University of Alabama Box 870350 Tuscaloosa, Alabama 35487-0350 U.S.A., e-mail: [email protected]
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Abstract

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It is shown that under $\text{ZFC}$ almost all planar compacta that meet every line in at most two points are subsets of sets that meet every line in exactly two points. This result was previously obtained by the author jointly with K. Kunen and J. vanMill under the assumption that Martin’s Axiom is valid.

Keywords

57N05
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 42 , Issue 1 , 01 March 1999 , pp. 46 - 51
Copyright
Copyright © Canadian Mathematical Society 1999

References

[1] Dijkstra, J. J., Kunen, K., and van Mill, J., Hausdorff measures and two point set extensions. Fund. Math. 157 (1998), 43–60.Google Scholar
[2] Dijkstra, J. J. and van Mill, J., Two point set extensions—a counterexample. Proc. Amer.Math. Soc. 125 (1997), 2501–2502.Google Scholar
[3] Mazurkiewicz, S., O pewnej mnogósci płaskiej, która ma z każda prosta dwa i tylko dwa punkty wspólne. C. R. Varsovie 7 (1914), 382–384; Polish; French transl. Sur un ensemble plan qui a avec chaque droite deux et seulement deux points communs. Stefan Mazurkiewicz, Traveaux de Topologie et ses Applications, PWN, Warsaw, 1969, 46–47.Google Scholar