The onto mapping property of Sierpinski

Define () There exists $({\varphi }_n:{\omega }_1\to {\omega }_1:n<\omega )$ such that for every uncountable $I$ which is a subset of ${\omega }_1$ there exists $n$ such that ${\varphi }_n$ 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 (*).