### TOPOLOGIES RELATED TO SETS HAVING THE BAIRE PROPERTY

Ewa Lazarow, Roy A. Johnson, Władysław Wiłczyński
1989 Demonstratio Mathematica
TOPOLOGIES RELATED TO SETS HAVING THE BAIRE PROPERTY S4 Scheinberg ¿1, [71 hae i.^'-ce tructsd thras topologies for the reals auch that the Borel set;* gererated by these topologies are exactly the Lsbeegue ma »¡surafel.*» Bets, Here we shall dj tha same for sets having the Bairs pro>v Our general construction of a topology will ba bas«il upon a lower density having sosjo additional, properties, this -*ill V/s tha content of tha first part In i << aocond part we shall gi?9 three examples of
more » ... r cansity leading to tiirs? differeci topologies which can be considered as category analogues of topologies dp.Escribsd in [?]» It should be mentioned that Scheinberg did not-use tue languor.of lower density, but hie results can be aaeiiy trp.na.la"'^ci > We sh«ll a 7: amine basic properties of all thre« top x-.ogiea, noma of the«, which depend only on the fact that the topology J.a described by lower density, are presented iu part t the r^si depending on the form of the lower density -in parx 2, In the sequel R will denote the real line, 33-the family of subsets of R having the Baire property, 3 -the 6-ideal of sets of the first category, N -tha set of natural numbers. Ve shall say that set» A,B e JJ are equivalent (A~B) if and only if AAB e d. Except where a topology J is specifically mentioned, all topological notions are with respect to the natural topology on R. So if 7 is some topology, then the notations T-open, 7-Sorel, T-Int, J-Cl and so on are self explaining. Recall that 6 is regular open if and only if G«Int(Cl G). -179 -