Spaces which contain a copy of the rationals
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.
scientific article; zbMATH DE number 4166033