# 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

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topDijkstra, 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

top- [1] J. J. Dijkstra, Generic partial two-point sets are extendable, Canad. Math. Bull., to appear. Zbl1002.54004
- [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] J. Kulesza, A two-point set must be zero-dimensional, ibid. 116 (1992), 551-553. Zbl0765.54006
- [4] N. S. Landkof, Foundations of Modern Potential Theory, Grundlehren Math. Wiss. 180, Springer, Berlin, 1972. Zbl0253.31001
- [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. Zbl0931.28001
- [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. Zbl0811.54002
- [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. Zbl0925.00005
- [12] M. Tsuji, Potential Theory in Modern Function Theory, Maruzen, Tokyo, 1959. Zbl0087.28401

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