×

Small sets with respect to certain classes of topologies. (English) Zbl 0917.54044

Let \(E\) be a set and \(\Gamma\) a group of transformations of \(E\). Consider the class \({\mathcal T}(E,\Gamma)\) of all topologies on \(E\) that are Baire space topologies with ccc whose classes of first category sets and sets with the Baire property are \(\Gamma\)-invariant. A set \(X\subset E\) is said to be a first category set with respect to \({\mathcal T}(E,\Gamma)\) if for any topology \(T\in{\mathcal T}(E,\Gamma)\) there is a stronger topology \(T'\in{\mathcal T}(E,\Gamma)\) such that every set that is first category with respect to \(T\) is first category with respect to \(T'\), and \(X\) is a first category set with respect to \(T'\).
The paper contains a characterization of first category sets with respect to \({\mathcal T}(E,\Gamma)\) in terms of orbits of \(\Gamma\) and several examples of pairs \((E,\Gamma)\) such that \(E\) admits a countable covering by sets that are first category sets with respect to \({\mathcal T}(E,\Gamma)\); in particular, it is shown that this is the case when \(E\) is a normed vector space and \(\Gamma\) is a non-separable additive subgroup of \(E\), and when \(E\) is the real line and \(\Gamma\) is an uncountable additive subgroup of \(E\). Some corollaries related to proper extensions of \(\Gamma\)-quasi-invariant measures and topologies are obtained.

MSC:

54H05 Descriptive set theory (topological aspects of Borel, analytic, projective, etc. sets)
28A05 Classes of sets (Borel fields, \(\sigma\)-rings, etc.), measurable sets, Suslin sets, analytic sets
54A10 Several topologies on one set (change of topology, comparison of topologies, lattices of topologies)
54E52 Baire category, Baire spaces
PDFBibTeX XMLCite
Full Text: DOI EuDML