Covering Property Axiom CPA$_{\rm cube}$ and its consequences

Krzysztof Ciesielski¹ ; Janusz Pawlikowski²

¹ Department of Mathematics West Virginia University Morgantown, WV 26506-6310, U.S.A.
² Department of Mathematics University of Wrocław Pl. Grunwaldzki 2/4 50-384 Wrocław, Poland

Fundamenta Mathematicae, Tome 176 (2003) no. 1, pp. 63-75.

Résumé

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

Détail

DOI : 10.4064/fm176-1-5

Keywords: formulate covering property axiom cpa cube which holds iterated perfect set model implies easily following facts every subset mathbb cardinality continuum there exists uniformly continuous function colon mathbb mathbb subset mathbb either perfectly meager universally null has cardinality frak cof cal omega frak cofinality measure ideal cal omega every uniformly bounded sequence langle mathbb mathbb rangle omega borel functions there sequences langle subset mathbb colon omega rangle compact sets langle omega omega colon omega rangle mathbb bigcup omega every omega langle upharpoonright rangle monotone uniformly convergent sequence uniformly continuous functions total failure martins axiom frak omega every non trivial ccc forcing mathbb there exist omega dense sets mathbb filter intersects them