Rose, D. A.; Janković, D.; Hamlett, T. R. Lower density topologies. (English) Zbl 0815.28001 Andima, Susan (ed.) et al., Papers on general topology and applications. Proceedings of the seventh summer conference in honor of Mary Ellen Rudin and her work, held at the University of Wisconsin, Madison, WI, USA, June 26 to 29, 1991. New York, NY: The New York Academy of Sciences,. Ann. N. Y. Acad. Sci. 704, 309-321 (1993). Summary: By considering lower density operators and their induced topologies in a general setting, some results of S. Scheinberg and E. Łazarow et al. are unified and generalized. It is also shown that every \(\sigma\)-finite complete measure space \((X, {\mathcal M}, m)\) has a lower density operator and that every such operator induces a topology making \(X\) a category measure space in the sense of J. C. Oxtoby, except that the measure need not be finite. One consequence is that category \(\sigma\)-finite measure spaces must have the countable chain condition. Also, for every topological space \((X, \tau)\), there is a lower density operator on the \(\sigma\)-field of sets having the property of Baire (relative to the \(\sigma\)-ideal of meager sets). Further, in both the “measure” and “category” contexts, all induced lower density topologies have simple form. Finally, it is shown that the deep \({\mathcal F}\)-density operator on the \(\sigma\)-field of subsets of the real line having the property of Baire is not a lower density operator.For the entire collection see [Zbl 0801.00034]. Cited in 3 Documents 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 54A10 Several topologies on one set (change of topology, comparison of topologies, lattices of topologies) Keywords:lower density operators; \(\sigma\)-finite complete measure space; category measure space; countable chain condition; lower density topologies PDFBibTeX XMLCite \textit{D. A. Rose} et al., in: Papers on general topology and applications. Proceedings of the seventh summer conference in honor of Mary Ellen Rudin and her work, held at the University of Wisconsin, Madison, WI, USA, June 26 to 29, 1991. New York, NY: The New York Academy of Sciences. 309--321 (1993; Zbl 0815.28001)