Une remarque sur les \(G_ \delta\)-denses
type: Item
instance of: scholarly article
title: Une remarque sur les \(G_ \delta\)-denses
zbMATH Open document ID: 0792.54013
DOI: 10.1007/BF01196432
author: Kamil Alnıaçık
author: Éric Saïas
published in: Archiv der Mathematik
publication date: 1994-08-01T00:00:00Z
review text: Let \(G\) be a dense \(G_ \delta\) subset of \(\mathbb{R}\) and \((f_ n)_{n\geq 0}\) a sequence of functions from \(\mathbb{R}\) to \(\mathbb{R}\), continuous and nowhere locally constant. By using Baire's theorem, the authors give an elementary proof that \(\bigcap_{n\geq 0} f^{-1}_ n(G)\) is also a \(G_ \delta\)-dense subset of \(\mathbb{R}\). By doing so, they generalize previous results of \textit{P. Erdős} [Michigan Math. J. 9, 59-60 (1962; Zbl 0114.26306)], \textit{G. J. Rieger} [Arch. Math. 26, 40-43 (1975; Zbl 0301.10029)] and \textit{W. Schwarz} [Math.-phys. Semesterber., n. F. 24, 84-87 (1977; Zbl 0344.10016)] on Liouville numbers.
Mathematics Subject Classification ID: 11J04
zbMATH DE Number: 590071
zbMATH Keywords: dense \(G_ \delta\) subset
zbMATH Keywords: Baire's theorem
zbMATH Keywords: \(G_ \delta\)-dense subset
MaRDI profile type: Publication
cites work: Representations of real numbers as sums and products of Liouville numbers
cites work: Über die Lösbarkeit von Gleichungssystemen durch Liouville-Zahlen
cites work: Q5516901
cites work: Q4111659
full work available at URL: https://doi.org/10.1007/bf01196432
OpenAlex ID: W2008864929
description: scientific article; zbMATH DE number 590071