Hausdorff measures and two point set extensions

• Autorzy: Jan J. Dijkstra , Kenneth Kunen , Jan van Mill
• Języki publikacji: EN

Abstrakty

EN

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.

Kategorie tematyczne

• 54G99: None of the above, but in this section
• 28A78: Hausdorff and packing measures

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Fundamenta Mathematicae
• Rocznik: 1998
• Tom: 157
• Numer: 1
• Strony: 43-60

Daty

wydano

1998

otrzymano

1997-06-17

poprawiono

1998-01-28

Twórcy

autor

Jan J. Dijkstra

• Department of Mathematics, The University of Alabama, Box 870350, Tuscaloosa, Alabama 35487-0350, U.S.A.

autor

Kenneth Kunen

• Faculteit Wiskunde en Informatica, Vrije Universiteit, De Boelelaan 1081a, 1081 HV Amsterdam, The Netherlands

autor

Jan van Mill

• Department of Mathematics, University of Wisconsin, Madison, Wisconsin 53706, U.S.A.

Bibliografia

• [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.
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• [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.
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