Journal of Applied Analysis 15 (2009), No. 1, 119--127
Copyright Heldermann Verlag 2009
When an Atomic and Complete Algebra of Sets is a Field of Sets with Nowhere Dense Boundary
Artur Bartoszewicz
Institute of Mathematics, Lódz Technical University, Wólczanska 215, 93-005 Lódz, Poland
arturbar@p.lodz.pl
Piotr Koszmider
Institute of Mathematics, Lódz Technical University, Wólczanska 215, 93-005 Lódz, Poland
piotr.koszmider@gmail.com
[Abstract-pdf]
We consider pairs $\langle{\cal A}, {\cal H}({\cal A})\rangle$ where ${\cal A}$ is an algebra of sets from some class called the class of algebras of type $\langle\kappa ,\lambda\rangle$ and where ${\cal H}({\cal A})$ is the ideal of hereditary sets of ${\cal A}$. We characterize which of the above pairs are topological, that is, which are fields of sets with nowhere dense boundary for some topology together with the ideal of nowhere dense sets for this topology. Making use of a theorem of Fichtenholz and Kantorovich which says that in ${\cal P}(\kappa)$ there is an independent family of cardinality $2^\kappa$, we construct an example of a pair $\langle$algebra, ideal $\rangle$ with complete quotient algebra and the hull property but not topological. This counterxample, given in ZFC, provides the complete solution of a problem posed by M. Balcerzak, A. Bartoszewicz and K. Ciesielski [Real Anal. Exchange 29 (2003/03) 265--273]. Such an algebra was constructed earlier by the first author [Topology Appl. 149 (2005) 9--15] under some aditional set theoretic assumption.
Keywords: Inner MB-representation, isomorphism of Boolean algebras, algebra of sets with nowhere dense boundary, Balcar-Franek theorem.
MSC: 28A05, 06E05, 03E35
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