Spaces which contain a copy of the rationals
type: Item
instance of: scholarly article
title: Spaces which contain a copy of the rationals
zbMATH Open document ID: 0709.54012
author: Katsuya Eda
author: Shizuo Kamo
author: T. Nogura
published in: Journal of the Mathematical Society of Japan
publication date: 1990-01-01T00:00:00Z
review text: A function from a topological space is called nowhere dense if all point inverses have empty interior. The authors show, in a key lemma, that images of the rationals \({\mathbb{Q}}\) under continuous nowhere dense functions contain a copy of \({\mathbb{Q}}\). This result is used to show that non-scattered Lashnev spaces and sequential spaces with reasonably small local characters and without isolated points contain a copy of \({\mathbb{Q}}\). Under CH, an elaborate proof of the existence of a non-scattered Fréchet space without isolated points that does not contain a copy of \({\mathbb{Q}}\) is also given. The paper also discusses closed embeddings of \({\mathbb{Q}}\) in product spaces. It is shown that unless a coordinate space itself has a copy of \({\mathbb{Q}}\), countable products and (under MA) even products of \(\kappa\)-many \((\kappa <c)\) spaces do not provide sufficient room for a closed embedding of \({\mathbb{Q}}\). All the results are interesting.
Mathematics Subject Classification ID: 54C25
Mathematics Subject Classification ID: 54F65
Mathematics Subject Classification ID: 54A25
zbMATH DE Number: 4166033
zbMATH Keywords: embedding the rationals
zbMATH Keywords: nowhere dense mapping
zbMATH Keywords: Cohen extension
zbMATH Keywords: non- scattered Lashnev spaces
zbMATH Keywords: sequential spaces
zbMATH Keywords: closed embeddings
zbMATH Keywords: product spaces
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.2969/jmsj/04210103
OpenAlex ID: W2047799893
DOI: 10.2969/JMSJ/04210103
description: scientific article; zbMATH DE number 4166033