A Jones space that is not countable dense homogeneous
type: Item
instance of: scholarly article
title: A Jones space that is not countable dense homogeneous
zbMATH Open document ID: 0834.54011
DOI: 10.1016/0166-8641(93)90105-M
author: W. L. Saltsman
author: Robert W. Heath
published in: Topology and its Applications
publication date: 1996-04-01T00:00:00Z
review text: A separable space \(X\) is countable dense homogeneous (abbreviated CDH) if for all countable dense subsets \(A\) and \(B\) of \(X\) there exists a homeomorphism \(h: X\to X\) with \(h(A)= B\). A space \(X\) is strongly locally homogeneous (SLH) if it has a base for the open sets \({\mathcal B}\) such that for every \(B\in {\mathcal B}\) and points \(x,y\in B\) there exists a homeomorphism \(f: X\to X\) with \(f(x)= y\) and \(f(z)= z\) for \(z\not\in B\). It is known that SLH Polish spaces are CDH. The reviewer showed that connected Baire SLH spaces need not be CDH [Proc. Am. Math. Soc. 84, 143- 148 (1982; Zbl 0486.54012)]\ and asked for examples of CDH connected spaces that are not SLH. In this interesting paper, the authors look for an example in the class of Jones spaces. A Jones space is the graph of a possibly discontinuous, additive function from the reals into itself. They prove that no Jones space is SLH and that there is a Jones space that is not CDH.
Mathematics Subject Classification ID: 54C99
Mathematics Subject Classification ID: 54G20
zbMATH DE Number: 474694
zbMATH Keywords: countable dense homogeneous
zbMATH Keywords: Jones space
reviewed by: Jan van Mill
MaRDI profile type: Publication
cites work: Countable dense homogeneous spaces
cites work: Q3833238
cites work: Conditions under which a connected representable space is locally connected
cites work: Some nonmetric, first countable, cancellative topological semigroups that are generalized metric spaces
cites work: Connected and disconnected plane sets and the functional equation 𝑓(𝑥)+𝑓(𝑦)=𝑓(𝑥+𝑦)
cites work: Strong Local Homogeneity Does not Imply Countable Dense Homogeneity
cites work: Q4030161
full work available at URL: 0166-8641(93)90105-m
OpenAlex ID: W2012844357
description: scientific article; zbMATH DE number 474694