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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Homeomorphisms of two-point sets


Authors: Ben Chad and Chris Good
Journal: Proc. Amer. Math. Soc. 139 (2011), 2287-2293
MSC (2010): Primary 05A99, 51N99, 54G99, 54H15
Published electronically: March 7, 2011
MathSciNet review: 2784793
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Abstract: Given a cardinal $ \kappa \leq \ensuremath{\mathfrak{c}}$, a subset of the plane is said to be a $ \kappa$-point set if and only if it meets every line in precisely $ \kappa$ many points. In response to a question of Cobb, we show that for all $ 2 \leq \kappa, \lambda < \ensuremath{\mathfrak{c}}$ there exists a $ \kappa$-point set which is homeomorphic to a $ \lambda$-point set, and further, we also show that it is consistent with ZFC that for all $ 2 \leq \kappa < \ensuremath{\mathfrak{c}}$, there exists a $ \kappa$-point set $ X$ such that for all $ 2 \leq \lambda < \ensuremath{\mathfrak{c}}$, $ X$ is homeomorphic to a $ \lambda$-point set. On the other hand, we prove that it is consistent with ZFC that for all $ 2 \leq \kappa, \lambda < \ensuremath{\mathfrak{c}}$, there exists a $ \kappa$-point set, such that for all homeomorphisms $ f:\mathbb{R}^2\rightarrow\mathbb{R}^2$, if $ f(X)$ is a $ \lambda$-point set, then $ \lambda = \kappa$.


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Additional Information

Ben Chad
Affiliation: St Edmund Hall, University of Oxford, Oxford, OX1 4AR, United Kingdom
Email: chad@maths.ox.ac.uk

Chris Good
Affiliation: School of Mathematics and Statistics, The University of Birmingham, Edgbaston, Birmingham, B15 2TT, United Kingdom
Email: c.good@bham.ac.uk

DOI: http://dx.doi.org/10.1090/S0002-9939-2011-10606-3
PII: S 0002-9939(2011)10606-3
Received by editor(s): October 2, 2009
Received by editor(s) in revised form: March 31, 2010
Published electronically: March 7, 2011
Additional Notes: The authors thank Rolf Suabedissen for his helpful suggestion, which led to our main result in Section \ref{universal}. They thank Robin Knight for his advice concerning properties of $\ensuremath{\mathfrak{c}}$. And they thank an anonymous referee for helpful comments concerning an earlier draft of this paper.
Communicated by: Alexander N. Dranishnikov
Article copyright: © Copyright 2011 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.