×

Generalization of the Scheeffer’s theorem. (English) Zbl 1037.54500

Summary: There is an interesting theorem in point set theory, the Scheeffer’s theorem, which says that if \(G\) and \(A\) are a nowhere dense subset and a countable subset of \(\mathbb{R}\) respectively, then for any given interval \((a,b)\), there exists \(c\in(a,b)\) such that \(A+c\) does not intersect \(G\). The aim of this paper is to generalize Scheeffer’s theorem in two aspects: generalize \(G\) to be a set of first category; generalize \(R\) to be the so-called translation space.

MSC:

54B05 Subspaces in general topology
54E52 Baire category, Baire spaces
54E35 Metric spaces, metrizability
03E15 Descriptive set theory
PDFBibTeX XMLCite