## 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:

 3e+15 Descriptive set theory 3e+35 Consistency and independence results 3e+50 Continuum hypothesis and Martin’s axiom
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