Hausdorff measures and two point set extensions

Jan Dijkstra; Kenneth Kunen; Jan van Mill

Fundamenta Mathematicae (1998)

  • Volume: 157, Issue: 1, page 43-60
  • ISSN: 0016-2736

Abstract

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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.

How to cite

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Dijkstra, Jan, Kunen, Kenneth, and van Mill, Jan. "Hausdorff measures and two point set extensions." Fundamenta Mathematicae 157.1 (1998): 43-60. <http://eudml.org/doc/212277>.

@article{Dijkstra1998,
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.},
author = {Dijkstra, Jan, Kunen, Kenneth, van Mill, Jan},
journal = {Fundamenta Mathematicae},
keywords = {two point set; partial two point set; extendable partial two point set; Hausdorff dimension zero; logarithmic capacity zero; Lebesgue measure zero},
language = {eng},
number = {1},
pages = {43-60},
title = {Hausdorff measures and two point set extensions},
url = {http://eudml.org/doc/212277},
volume = {157},
year = {1998},
}

TY - JOUR
AU - Dijkstra, Jan
AU - Kunen, Kenneth
AU - van Mill, Jan
TI - Hausdorff measures and two point set extensions
JO - Fundamenta Mathematicae
PY - 1998
VL - 157
IS - 1
SP - 43
EP - 60
AB - 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.
LA - eng
KW - two point set; partial two point set; extendable partial two point set; Hausdorff dimension zero; logarithmic capacity zero; Lebesgue measure zero
UR - http://eudml.org/doc/212277
ER -

References

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  1. [1] J. J. Dijkstra, Generic partial two-point sets are extendable, Canad. Math. Bull., to appear. Zbl1002.54004
  2. [2] J. J. Dijkstra and J. van Mill, Two point set extensions - a counterexample, Proc. Amer. Math. Soc. 125 (1997), 2501-2502. Zbl0880.54022
  3. [3] J. Kulesza, A two-point set must be zero-dimensional, ibid. 116 (1992), 551-553. Zbl0765.54006
  4. [4] N. S. Landkof, Foundations of Modern Potential Theory, Grundlehren Math. Wiss. 180, Springer, Berlin, 1972. Zbl0253.31001
  5. [5] S. Lang, Algebra, 3rd ed., Addison-Wesley, Reading, 1993. 
  6. [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. [7] R. D. Mauldin, On sets which meet each line in exactly two points, Bull. London Math. Soc., to appear. Zbl0931.28001
  8. [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. [9] J. van Mill and G. M. Reed, Open problems in topology, Topology Appl. 62 (1995), 93-99. Zbl0811.54002
  10. [10] M. E. Rudin, Martin's Axiom, in: Handbook of Mathematical Logic, North-Holland, Amsterdam, 1977, 491-501. 
  11. [11] W. Rudin, Real and Complex Analysis, 3rd ed., McGraw-Hill, New York, 1987. Zbl0925.00005
  12. [12] M. Tsuji, Potential Theory in Modern Function Theory, Maruzen, Tokyo, 1959. Zbl0087.28401

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