Real Analysis Exchange

Marczewski Fields and Ideals

Abstract

For an $X\neq\emptyset$ and a given family $\mathcal{F}\subset \mathcal {P}(X)\setminus \{ \emptyset \}$, we consider the \mf \sodf which consists of sets $A\subset X$ such that each set $U\in \mathcal{F}$ contains a set $V\in \mathcal{F}$ with $V\subset A$ or $V\cap A=\emptyset$. We also study the respective ideal $S^0(\mathcal{F})$. We show general properties of $S^0(\mathcal{F})$ and certain representation theorems. For instance we prove that the interval algebra in $[0,1)$ is a Marczewski field. We are also interested in situations where $S(\mathcal{F}=S(\tau\setminus \{ \emptyset \} )$ for a topology $\tau$ on $X$. We propose a general method which establishes $S(\mathcal{F})$ and $S^0(\mathcal{F})$ provided that $\mathcal{F}$ is the family of perfect sets with respect to $\tau$, and $\tau$ is a certain ideal topology on $\mathbb{R}$ connected with measure or category.

Article information

Source
Real Anal. Exchange, Volume 26, Number 2 (2000), 703-716.

Dates
First available in Project Euclid: 27 June 2008

https://projecteuclid.org/euclid.rae/1214571361

Mathematical Reviews number (MathSciNet)
MR1844387

Zentralblatt MATH identifier
1009.28001

Citation

Balcerzak, M.; Bartoszewicz, A. Marczewski Fields and Ideals. Real Anal. Exchange 26 (2000), no. 2, 703--716. https://projecteuclid.org/euclid.rae/1214571361

References

• M. Balcerzak, A. Bartoszewicz, K. Ciesielski, On Marczewski-Burstin representations of certain algebras of sets, Real Anal. Exchange, 26 (2000-2001).
• M. Balcerzak, J. Rzepecka, \m sets in the \h topologies for measure and category, Acta Univ. Carolin. Math. Phys., 39 (1998), 93–97.
• J. Brendle, Strolling through paradise, \fm 148 (1995), 1–25.
• J. B. Brown, The \ra sets and related sigma algebras and ideals, the lecture during Special Session on Measure Theory and Descriptive Set Theory, Atlanta Meeting of the AMS, January 1988.
• J. B. Brown, The \ra sets and related sigma algebras and ideals, \fm 136 (1990), 179–185.
• J. B. Brown, G. V. Cox, Classical theory of totally imperfect spaces, \rae 7 (1982), 1–39.
• J. B. Brown, H. Elalaoui-Talibi, Marczewski-Burstin-like characterizations of $\sigma$-algebras, ideals, and measurable functions, Colloq. Math., 82 (1999), 277–286.
• C. Burstin, Eigenschaften messbaren und nichtmessbaren Mengen, Wien Ber. 123 (1914), 1525–1551.
• K. Ciesielski, J. Jasinski, Topologies making a given ideal nowhere dense or meager, Topology Appl. 63 (1995), 277–298.
• K. Ciesielski, L. Larson, K. Ostaszewski, ${\cal I}$-density continuous functions, Mem. Amer. Math. Soc. 107 (515) (1994).
• P. Corazza, Ramsey sets, the Ramsey ideal, and other classes over R, J. Symb. Logic 57 (1992), 1441–1468.
• F. J. Freniche, The number of nonisomorphic Boolean subalgebras of a power set, Proc. Amer. Math. Soc. 91 (1984), 199–201.
• H. Hashimoto, On the $^\star$topology and its application, \fm 91 (1976), 5–10.
• D. Jankovič and T. R. Hamlett, New topologies from old via ideals, Amer. Math. Monthly, 97, No 4, April 1990, 295–310.
• S. Kopelberg, Handbook of Boolean Algebras, vol.1, North Holland, 1989.
• K. Kuratowski, Topology, vol.1, Academic Press, New York, 1966.
• J. Lukeš, J. Malý and L. Zajiček, Fine Topology Methods in Real Analysis and Potential Theory, Lecture Notes in Math. 1189, Springer, New York, 1986.
• E. Łazarow, R. A. Johnson, W. Wilczyński, Topologies related to sets having the Baire property, Demonstr. Math. 22 (1989), 179–191.
• A. W. Miller, Special subsets of the real line, in Handbook of Set-Theoretic Topology, North-Holland, Amsterdam-New York, 1984, 201–233.
• A. W. Miller, Special sets of reals, Israel Math. Conf. Proc., 6 (1993), 415–431.
• J. C. Morgan II, Point Set Theory, Marcel Dekker, New York, 1990.
• J. C. Oxtoby, Measure and Category, Springer Verlag, New York 1971.
• J. Pawlikowski, Parametrized Ellentuck theorem, Topology Appl. 37 (1990), 65–73.
• W. Poreda, E. Wagner-Bojakowska, W. Wilczyński, A category analogue of the density topology, \fm 125 (1985), 167–173.
• P. Reardon, Ramsey, \leb and \m sets and the Baire property, \fm 149 (1996), 191–203.
• E. Szpilrajn (Marczewski), Sur une classe de fonctions de M. Sierpiński et la classe correspondante d'ensembles, \fm 24 (1935), 17–34.
• J. T. Walsh, Marczewski sets, measure and the Baire property, Fund. Math. 129 (1988), 83–89.
• W. Wilczyński, A generalization of the density topology, \rae 8 (1982-83), 16–20.
• W. Wilczyński, A category analogue of the density topology, approximate continuity and the approximate derivative, \rae 10 (1984-85), 241–265.
• S. Wroński, On fields of sets with a nowhere dense boundary, Demonstr. Math. 12 (1979), 373–377.
• S. Wroński, On proper subuniverses of a Boolean algebra, Acta Univ. Lodziensis, Folia Mathematica 9 (1997), 69–76.