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ArticleOriginal scientific text

Title

Hausdorff measures and two point set extensions

Authors 1, 2, 3

Affiliations

  1. Department of Mathematics, The University of Alabama, Box 870350, Tuscaloosa, Alabama 35487-0350, U.S.A.
  2. Faculteit Wiskunde en Informatica, Vrije Universiteit, De Boelelaan 1081a, 1081 HV Amsterdam, The Netherlands
  3. Department of Mathematics, University of Wisconsin, Madison, Wisconsin 53706, U.S.A.

Abstract

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.

Bibliography

  1. J. J. Dijkstra, Generic partial two-point sets are extendable, Canad. Math. Bull., to appear.
  2. J. J. Dijkstra and J. van Mill, Two point set extensions - a counterexample, Proc. Amer. Math. Soc. 125 (1997), 2501-2502.
  3. J. Kulesza, A two-point set must be zero-dimensional, ibid. 116 (1992), 551-553.
  4. N. S. Landkof, Foundations of Modern Potential Theory, Grundlehren Math. Wiss. 180, Springer, Berlin, 1972.
  5. S. Lang, Algebra, 3rd ed., Addison-Wesley, Reading, 1993.
  6. R. D. Mauldin, Problems in topology arising from analysis, in: Open Problems in Topology, J. van Mill and G. M. Reed (eds.), North-Holland, Amsterdam, 1990, 617-629.
  7. R. D. Mauldin, On sets which meet each line in exactly two points, Bull. London Math. Soc., to appear.
  8. S. Mazurkiewicz, O pewnej mnogości płaskiej, która ma każdą prostą dwa i tylko dwa punkty wspólne, C. R. Varsovie 7 (1914), 382-384 (in Polish); French transl.: Sur un ensemble plan qui a avec chaque droite deux et seulement deux points communs, in: Stefan Mazurkiewicz, Traveaux de Topologie et ses Applications, PWN, Warszawa, 1969, 46-47.
  9. J. van Mill and G. M. Reed, Open problems in topology, Topology Appl. 62 (1995), 93-99.
  10. M. E. Rudin, Martin's Axiom, in: Handbook of Mathematical Logic, North-Holland, Amsterdam, 1977, 491-501.
  11. W. Rudin, Real and Complex Analysis, 3rd ed., McGraw-Hill, New York, 1987.
  12. M. Tsuji, Potential Theory in Modern Function Theory, Maruzen, Tokyo, 1959.
Fundamenta Mathematicae
1998/157/1
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Pages:
43-60
Main language of publication
English
Received
1997-06-17
Accepted
1998-01-28
Published
1998
Exact and natural sciences