The onto mapping property of Sierpinski

Abstract

Define (*) There exists $(\phi_n:\omega_1\to \omega_1:n<\omega)$ such that for every uncountable $I$ which is a subset of $\omega_1$ there exists $n$ such that $\phi_n$ maps $I$ onto $\omega_1$. This is roughly what Sierpinski in his book on the continuum hypothesis refers to as $P_3$ but I think he brings reals number line into it. I don't know French so I cannot say for sure what he says but I think he proves that (*) follows from the continuum hypothesis. We show that the existence of a Luzin set implies (*); and (*) implies that there exists a nonmeager set of reals of size $\omega_1$. We also show that it is relatively consistent that (*) holds but there is no Luzin set. All the other properties in this paper, (**), (S*), (S**), (B*) are shown to be equivalent to (*).