  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 (*).
