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Magic sets. (English) Zbl 06924880

Summary: 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 \(f,g\mathcal{H}\) 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 \(\lim_{h\downarrow 0}\frac{1}{2}(f(x + h) + f(x - h)) = f(x)\). 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.

MSC:

03E15 Descriptive set theory
03E35 Consistency and independence results
03E50 Continuum hypothesis and Martin’s axiom
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