Property C'', strong measure zero sets and subsets of the plane

Janusz Pawlikowski

Fundamenta Mathematicae (1997)

  • Volume: 153, Issue: 3, page 277-293
  • ISSN: 0016-2736

Abstract

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Let X be a set of reals. We show that  • X has property C" of Rothberger iff for all closed F ⊆ ℝ × ℝ with vertical sections F x (x ∈ X) null, ∪ x ∈ X F x is null;  • X has strong measure zero iff for all closed F ⊆ ℝ × ℝ with all vertical sections F x (x ∈ ℝ) null, ∪ x ∈ X F x is null.

How to cite

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Pawlikowski, Janusz. "Property C'', strong measure zero sets and subsets of the plane." Fundamenta Mathematicae 153.3 (1997): 277-293. <http://eudml.org/doc/212224>.

@article{Pawlikowski1997,
abstract = {Let X be a set of reals. We show that  • X has property C" of Rothberger iff for all closed F ⊆ ℝ × ℝ with vertical sections $F_x$ (x ∈ X) null, $∪_\{x ∈ X\}F_x$ is null;  • X has strong measure zero iff for all closed F ⊆ ℝ × ℝ with all vertical sections $F_x$ (x ∈ ℝ) null, $∪_\{x ∈ X\}F_x$ is null.},
author = {Pawlikowski, Janusz},
journal = {Fundamenta Mathematicae},
keywords = {Rothberger property; strong measure zero},
language = {eng},
number = {3},
pages = {277-293},
title = {Property C'', strong measure zero sets and subsets of the plane},
url = {http://eudml.org/doc/212224},
volume = {153},
year = {1997},
}

TY - JOUR
AU - Pawlikowski, Janusz
TI - Property C'', strong measure zero sets and subsets of the plane
JO - Fundamenta Mathematicae
PY - 1997
VL - 153
IS - 3
SP - 277
EP - 293
AB - Let X be a set of reals. We show that  • X has property C" of Rothberger iff for all closed F ⊆ ℝ × ℝ with vertical sections $F_x$ (x ∈ X) null, $∪_{x ∈ X}F_x$ is null;  • X has strong measure zero iff for all closed F ⊆ ℝ × ℝ with all vertical sections $F_x$ (x ∈ ℝ) null, $∪_{x ∈ X}F_x$ is null.
LA - eng
KW - Rothberger property; strong measure zero
UR - http://eudml.org/doc/212224
ER -

References

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  1. [AR] A. Andryszczak and I. Recław, A note on strong measure zero sets, Acta Univ. Carolin. Math. Phys. 34 (2) (1993), 7-9. 
  2. [BS] T. Bartoszyński and S. Shelah, Closed measure zero sets, Ann. Pure Appl. Logic 58 (1992), 93-110. Zbl0764.03018
  3. [D] C. Dellacherie, Un cours sur les ensembles analytiques, in: Analytic Sets, C. A. Rogers et al. (eds.), Academic Press, 1980, 183-316. 
  4. [FM] D. H. Fremlin and A. Miller, On some properties of Hurewicz, Menger and Rothberger, Fund. Math. 129 (1988), 17-33. Zbl0665.54026
  5. [G] F. Galvin, Indeterminacy of point-open games, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys. 26 (1978), 445-449. Zbl0392.90101
  6. [GMS] F. Galvin, J. Mycielski and R. M. Solovay, Strong measure zero sets, Notices Amer. Math. Soc. 26 (1979), A-280. 
  7. [Ke] A. S. Kechris, Classical Descriptive Set Theory, Grad. Texts in Math. 156, Springer, 1995. 
  8. [M] A. W. Miller, Special subsets of the real line, in: Handbook of Set-Theoretic Topology, K. Kunen and J. E. Vaughan (eds.), North-Holland, 1984, 201-233. 
  9. [M1] A. W. Miller, Some properties of measure and category, Trans. Amer. Math. Soc. 266 (1981), 93-114, and 271 (1982), 347-348. Zbl0472.03040
  10. [P] J. Pawlikowski, Undetermined sets of point-open games, Fund. Math. 144 (1994), 279-285. Zbl0853.54033
  11. [P1] J. Pawlikowski, A characterization of strong measure zero sets, Israel J. Math. 93 (1996), 171-184. Zbl0857.28001
  12. [R] I. Recław, Every Lusin set is undetermined in the point-open game, Fund. Math. 144 (1994), 43-54. Zbl0809.04002
  13. [Sh] S. Shelah, Vive la différence I: Nonisomorphism of ultrapowers of countable models, in: Set Theory of the Continuum, H. Judah, W. Just and H. Woodin (eds.), Springer, 1992, 357-405. Zbl0789.03035
  14. [T] J. Truss, Sets having calibre ℵ 1 , in: Logic Colloquium ’76, R. Gandy and M. Hyland (eds.), Stud. Logic Found. Math. 87, North-Holland, 1977, 595-612. 

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