Model with no magic set

We will prove that there exists a model of ZFC+“c = ω2” in which every M ⊆ R of cardinality less than continuum c is meager, and such that for every X ⊆ R of cardinality c there exists a continuous function f : R → R with f [X] = [0, 1]. In particular in this model there is no magic set, i.e., a set M ⊆ R such that the equation f [M ] = g[M ] implies f = g for every continuous nowhere constant functions f, g : R → R. ∗The authors wish to thank Professors Maxim Burke, Andrzej RosC lanowski and Jerzy Wojciechowski for reading preliminary versions of this paper and helping in improving its final version. The work of the first author was partially supported by the NSF Cooperative Research Grant INT-9600548 and 1996/97 West Virginia University Senate Research Grant. †This work was supported in part by a grant from “Basic Research Foundation” of the Israel Academy of Sciences and Humanities. Publication 653.