×

Topologies related to sets having the Baire property. (English) Zbl 0692.28001

Let \({\mathcal B}\) be the family of subsets of the real line \({\mathbb{R}}\) having the Baire property, let \({\mathcal I}\) be the \(\sigma\)-ideal of sets of the first category. Two sets \(A,B\in {\mathcal B}\) are said to be equivalent, \(A\sim B,\) if the symmetric difference \(A\Delta\) B belongs to \({\mathcal I}\). An operation \(\Phi:{\mathcal B}\to 2^{{\mathbb{R}}}\) is said to be a category lower density if it fulfils the following conditions: 1. for each \(A\in {\mathcal B},\quad \Phi (A)\sim A;\) 2. if \(A\sim B,\) then \(\Phi (A)=\Phi (B);\quad 3.\quad \Phi (\emptyset)=\emptyset,\quad \Phi ({\mathbb{R}})={\mathbb{R}};\quad 4.\quad \Phi (A\cap B)=\Phi (A)\cap \Phi (B);\) 5. if \(A=G\Delta P,\) where G is regular open and \(P\in {\mathcal I},\) then \(G\subset \Phi (A)\subset Cl(G)\) (Cl(Z) - the closure of the set Z). The authors examine properties of the topology \({\mathcal T}=\{\Phi (A)-P:A\in {\mathcal B},P\in {\mathcal I}\}\) in general, and, using some special forms of the operation \(\Phi\), they show, e.g., the \({\mathcal T}\)-Borel sets are precisely the sets having the Baire property, each \({\mathcal T}\)-Borel set is a \({\mathcal T}-F_{\sigma \delta}\) set and there exists a set in \({\mathcal B}\) which is not a \({\mathcal T}-G_{\delta}\) set.
Reviewer: P.Kostyrko

MSC:

28A05 Classes of sets (Borel fields, \(\sigma\)-rings, etc.), measurable sets, Suslin sets, analytic sets
26A03 Foundations: limits and generalizations, elementary topology of the line
26A21 Classification of real functions; Baire classification of sets and functions
PDFBibTeX XMLCite