Real Analysis Exchange

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

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}$. Article information Source Real Anal. Exchange, Volume 29, Number 2 (2003), 755-762. Dates First available in Project Euclid: 7 June 2006 Permanent link to this document https://projecteuclid.org/euclid.rae/1149698537 Mathematical Reviews number (MathSciNet) MR2083810 Zentralblatt MATH identifier 1065.03028 Citation Ciesielski, Krzysztof; Pawlikowski, Janusz. Continuous images of big sets and additivity of$s_0$under cpa$_{prism}$. Real Anal. Exchange 29 (2003), no. 2, 755--762. https://projecteuclid.org/euclid.rae/1149698537 References • K. Ciesielski, Set Theory for the Working Mathematician, • K. Ciesielski, J. Pawlikowski, Crowded and selective ultrafilters under the Covering Property Axiom, J. Appl. Anal. 9(1) (2003), 19–55. (Preprint$^\star$available.)\footnotePreprints marked by$^\star$are available in electronic form from Set Theoretic Analysis Web Page: http://www.math.wvu.edu/homepages/kcies/STA/STA.html • K. Ciesielski, J. Pawlikowski, Small coverings with smooth functions under the Covering Property Axiom, preprint$^\star$. • K. Ciesielski, J. Pawlikowski, Covering Property Axiom CPA, version of January 2003, work in progress$^\star$. • K. Ciesielski, J. Pawlikowski, Covering Property Axiom CPA$_{\rm cube}$and its consequences, Fund. Math., 176(1) (2003), 63–75. (Preprint$^\star$available.) • H. Judah, A. W. Miller, S. Shelah, Sacks forcing, Laver forcing, and Martin's axiom, Arch. Math. Logic, 31(3) (1992), 145–161. • V. Kanovei, Non-Glimm–Effros equivalence relations at second projective level, Fund. Math., 154 (1997), 1–35. • A. W. Miller, Mapping a set of reals onto the reals, J. Symbolic Logic, 48 (1983), 575–584. • A. W. Miller, {Special Subsets of the Real • A. Nowik, Possibly there is no uniformly completely Ramsey null set of size$2^{\omega}$, Colloq. Math., 93 (2002), 251–258. (Preprint$^\star$available.) • P. Simon, Sacks forcing collapses$\continuum$to$\mathfrak b\$, Comment. Math. Univ. Carolin., 34(4) (1993), 707–710.
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