A model with no magic sets

by Ciesielski and Shelah. [CiSh:653]
J Symbolic Logic, 1999
We will prove that there exists a model of ZFC+``c = omega_2'' in which every M subseteq R of cardinality less than continuum c is meager, and such that for every X subseteq 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 subseteq R such that the equation f[M]=g[M] implies f=g for every continuous nowhere constant functions f,g:R-> R .

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