Real Analysis Exchange

Magic Sets

Lorenz Halbeisen, Marc Lischka, and Salome Schumacher

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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.

Article information

Source
Real Anal. Exchange, Volume 43, Number 1 (2018), 187-204.

Dates
First available in Project Euclid: 2 May 2018

Permanent link to this document
https://projecteuclid.org/euclid.rae/1525226429

Digital Object Identifier
doi:10.14321/realanalexch.43.1.0187

Mathematical Reviews number (MathSciNet)
MR3816438

Zentralblatt MATH identifier
06924880

Subjects
Primary: 03E20: Other classical set theory (including functions, relations, and set algebra)
Secondary: 26A99: None of the above, but in this section 03E50: Continuum hypothesis and Martin's axiom [See also 03E57] 03E35: Consistency and independence results

Keywords
magic sets symmetric functions Continuum Hypothesis Martin's Axiom sets of range uniqueness

Citation

Halbeisen, Lorenz; Lischka, Marc; Schumacher, Salome. Magic Sets. Real Anal. Exchange 43 (2018), no. 1, 187--204. doi:10.14321/realanalexch.43.1.0187. https://projecteuclid.org/euclid.rae/1525226429


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