Title: The onto mapping property of Sierpinski
Author: Arnold W. Miller
Date: July 2014
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 (*).
Latest version:
http://www.math.wisc.edu/~miller/res/sier.pdf