Continuous images of big sets and additivity of $s_0$ under cpa$_{prism}$.

Abstract

We prove that the Covering Property Axiom $CPA_{prism}, which holds in the iterated perfect set model, implies the following facts:

There exists a family $\mathcal{G}$ of uniformly continuous functions from $\mathbb{R}$ to $[0,1]$ such that $|\mathcal{G}|=\omega_1$ and for every $S\in[\mathbb{R}]^\mathfrak{c}$ there exists a $g\in\mathcal{G}$ with $g[S]=[0,1]$

The additivity of the Marczewski's ideal $s_0$ is equal to $\omega_1<\mathfrak{c}$.