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Abstract

We formulate a Covering Property Axiom ${\rm CPA}_{\rm cube}$, which holds in the iterated perfect set model, and show that it implies easily the following facts.

(a) For every $S\subset{\mathbb R}$ of cardinality continuum there exists a uniformly continuous function $g\colon\,{\mathbb R}\to{\mathbb R}$ with $g[S]=[0,1]$.

(b) If $S\subset{\mathbb R}$ is either perfectly meager or universally null then $S$ has cardinality less than~${\frak c}$.

(c) ${\rm cof}({\cal N})=\omega_1<{\frak c}$, i.e., the cofinality of the measure ideal ${\cal N}$ is $\omega_1$.

(d) For every uniformly bounded sequence $\langle f_n\in{\mathbb R}^{\mathbb R}\rangle_{n<\omega}$ of Borel functions there are sequences: $\langle P_\xi\subset{\mathbb R}\colon\,\xi<\omega_1\rangle$ of compact sets and $\langle W_\xi\in[\omega]^\omega\colon\,\xi<\omega_1\rangle$ such that ${\mathbb R}=\bigcup_{\xi<\omega_1}P_\xi$ and for every $\xi<\omega_1$, $\langle f_n\upharpoonright P_\xi\rangle_{n\in W_\xi}$ is a monotone uniformly convergent sequence of uniformly continuous functions.

(e) Total failure of Martin's Axiom: ${\frak c}>\omega_1$ and for every non-trivial ccc forcing ${\mathbb P}$ there exist $\omega_1$ dense sets in ${\mathbb P}$ such that no filter intersects all of them