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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 

 

A two-point set must be zero-dimensional


Author: John Kulesza
Journal: Proc. Amer. Math. Soc. 116 (1992), 551-553
MSC: Primary 54F45; Secondary 54B05, 54H05
DOI: https://doi.org/10.1090/S0002-9939-1992-1093599-1
MathSciNet review: 1093599
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Abstract: We prove that any subset of $ {\mathbb{R}^2}$ that intersects no line in more than two points either is zero-dimensional or contains an arc. This is used to prove that any subset of $ {\mathbb{R}^2}$ that intersects each line in exactly two points is zero-dimensional, answering a question of Mauldin.


References [Enhancements On Off] (What's this?)

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    Ryszard Engelking, Dimension theory, North-Holland Publishing Co., Amsterdam-Oxford-New York; PWN—Polish Scientific Publishers, Warsaw, 1978. Translated from the Polish and revised by the author; North-Holland Mathematical Library, 19. MR 0482697
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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1992-1093599-1
Keywords: Zero-dimensional, two-point set
Article copyright: © Copyright 1992 American Mathematical Society