Abstract

We will prove that there exists a model of $ZFC+"\mathfrak{c} = \omega_2"$ in which every $M \subseteq \mathbb{R}$ of cardinality less than continuum $\mathfrak{c}$ is meager, and such that for every $X \subseteq \mathbb{R}$ of cardinality $\mathfrak{c}$ there exists a continuous function $f: \mathbb{R} \rightarrow \mathbb{R}$ with f[X] = [0, 1]. In particular in this model there is no magic set, i.e., a set $M \subseteq \mathbb{R}$ such that the equation f[M] = g[M] implies f = g for every continuous nowhere constant functions $f, g: \mathbb{R} \rightarrow \mathbb{R}$.