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Proceedings of the American Mathematical Society

Published by the American Mathematical Society since 1950, Proceedings of the American Mathematical Society is devoted to shorter research articles in all areas of pure and applied mathematics.

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

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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Homeomorphisms of two-point sets
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by Ben Chad and Chris Good PDF
Proc. Amer. Math. Soc. 139 (2011), 2287-2293 Request permission

Abstract:

Given a cardinal $\kappa \leq \mathfrak {c}$, 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 $2 \leq \kappa , \lambda < \mathfrak {c}$ 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 $2 \leq \kappa < \mathfrak {c}$, there exists a $\kappa$-point set $X$ such that for all $2 \leq \lambda < \mathfrak {c}$, $X$ is homeomorphic to a $\lambda$-point set. On the other hand, we prove that it is consistent with ZFC that for all $2 \leq \kappa , \lambda < \mathfrak {c}$, 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
  • MR Author ID: 336197
  • ORCID: 0000-0001-8646-1462
  • Email: c.good@bham.ac.uk
  • 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 $\mathfrak {c}$. And they thank an anonymous referee for helpful comments concerning an earlier draft of this paper.
  • Communicated by: Alexander N. Dranishnikov
  • © Copyright 2011 American Mathematical Society
    The copyright for this article reverts to public domain 28 years after publication.
  • Journal: Proc. Amer. Math. Soc. 139 (2011), 2287-2293
  • MSC (2010): Primary 05A99, 51N99, 54G99, 54H15
  • DOI: https://doi.org/10.1090/S0002-9939-2011-10606-3
  • MathSciNet review: 2784793