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 $\lambda = \kappa$.References
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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