A Jones space that is not countable dense homogeneous
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.
scientific article; zbMATH DE number 474694