Real Analysis Exchange

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

Krzysztof Ciesielski and Janusz Pawlikowski

Full-text: Open access

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 $\omega_1<\mathfrak{c}$.

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

Subjects
Primary: 03E35: Consistency and independence results
Secondary: 03E17: Cardinal characteristics of the continuum 26A03: Foundations: limits and generalizations, elementary topology of the line

Keywords
Continuous images additivity Marczewski's ideal $s_0$.

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


Export citation

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.
  • J. Zapletal, Cardinal Invariants and Descriptive Set Theory, Mem. Amer. Math. Soc., to appear.