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2018 Magic Sets
Lorenz Halbeisen, Marc Lischka, Salome Schumacher
Real Anal. Exchange 43(1): 187-204 (2018). DOI: 10.14321/realanalexch.43.1.0187

Abstract

In this paper we study magic sets for certain families \(\mathcal{H}\subseteq {^\mathbb{R}\mathbb{R}}\) which are subsets \(M\subseteq\mathbb{R}\) such that for all functions \(f,g\in\mathcal{H}\) we have that \(g[M]\subseteq f[M]\Rightarrow f=g\). Specifically we are interested in magic sets for the family \(\mathcal{G}\) of all continuous functions that are not constant on any open subset of \(\mathbb{R}\). We will show that these magic sets are stable in the following sense: Adding and removing a countable set does not destroy the property of being a magic set. Moreover, if the union of less than \(\mathfrak{c}\) meager sets is still meager, we can also add and remove sets of cardinality less than \(\mathfrak{c}\) without destroying the magic set.

Then we will enlarge the family \(\mathcal{G}\) to a family \(\mathcal{F}\) by replacing the continuity with symmetry and assuming that the functions are locally bounded. A function \(f:\mathbb{R}\to\mathbb{R}\) is symmetric iff for every \(x\in\mathbb{R}\) we have that \(\lim_{h\downarrow 0}\frac{1}{2}\left(f(x+h)+f(x-h)\right )=f(x)\). For this family of functions we will construct \(2^\mathfrak{c}\) pairwise different magic sets which cannot be destroyed by adding and removing a set of cardinality less than \(\mathfrak{c}\). We will see that under the continuum hypothesis magic sets and these more stable magic sets for the family \(\mathcal{F}\) are the same. We shall also see that the assumption of local boundedness cannot be omitted. Finally, we will prove that for the existence of a magic set for the family \(\mathcal{F}\) it is sufficient to assume that the union of less than \(\mathfrak{c}\) meager sets is still meager. So for example Martin’s axiom for \(\sigma\)-centered partial orders implies the existence of a magic set.

Citation

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Lorenz Halbeisen. Marc Lischka. Salome Schumacher. "Magic Sets." Real Anal. Exchange 43 (1) 187 - 204, 2018. https://doi.org/10.14321/realanalexch.43.1.0187

Information

Published: 2018
First available in Project Euclid: 2 May 2018

zbMATH: 06924880
MathSciNet: MR3816438
Digital Object Identifier: 10.14321/realanalexch.43.1.0187

Subjects:
Primary: 03E20
Secondary: 03E35 , 03E50 , 26A99

Keywords: Continuum hypothesis , magic sets , Martin's Axiom , sets of range uniqueness , symmetric functions

Rights: Copyright © 2018 Michigan State University Press

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Vol.43 • No. 1 • 2018
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