The onto mapping property of Sierpinski

TL;DR

This paper explores the onto mapping property of the Sierpinski set, establishing its connections with Luzin sets, nonmeager sets, and various equivalent properties under different set-theoretic assumptions.

Contribution

It demonstrates that the onto mapping property is implied by the existence of Luzin sets and is consistent with models lacking Luzin sets, also establishing equivalences among related properties.

Findings

Luzin sets imply the onto mapping property (*)

(*) implies the existence of a nonmeager set of reals of size ω₁

(*) is consistent without the existence of Luzin sets

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 (*).