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Define  (*) There exists $(\phi_n:\omega_1\to \omega_1:n<\omega)$ such that for everyuncountable $I$ 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 hypothesisrefers to as $P_3$ but I think he brings reals number line into it. I don'tknow 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 aLuzin set implies (*); and (*) implies that there exists a nonmeager set ofreals 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 (*).

The onto mapping property of Sierpinski