1 results
THE ONTO MAPPING OF SIERPINSKI AND NONMEAGER SETS
-
- Journal:
- The Journal of Symbolic Logic / Volume 82 / Issue 3 / September 2017
- Published online by Cambridge University Press:
- 16 May 2017, pp. 958-965
- Print publication:
- September 2017
-
- Article
- Export citation
-
The principle (*) of Sierpinski is the assertion that there is a family of functions
$\left\{ {{\varphi _n}:{\omega _1} \to {\omega _1}|n \in \omega } \right\}$ such that for every 
$I \in {[{\omega _1}]^{{\omega _1}}}$ there is n ε ω such that 
${\varphi _n}[I] = {\omega _1}$. We prove that this principle holds if there is a nonmeager set of size ω1 answering question of Arnold W. Miller. Combining our result with a theorem of Miller it then follows that (*) is equivalent to 
$non\left( {\cal M} \right) = {\omega _1}$. Miller also proved that the principle of Sierpinki is equivalent to the existence of a weak version of a Luzin set, we will construct a model where all of these sets are meager yet 
$non\left( {\cal M} \right) = {\omega _1}$.
