Font Size: 

A space $X$ is countable dense homogeneous (CDH) if for all countable dense subsets $D$ and $E$ of $X$ there is a homeomorphism $f : X\to X$ such that $f(D) = E$. In this talk we are interested in CDH-subsets of the reals. The spaces we are interested in have no isolated points. No such countable space is CDH. It is known that every Borel CDH-subset of the reals is Polish. It is also known that there is a CDH-subset of the reals of cardinality $\aleph_1$. We show that sufficiently nice $\lambda$-sets are CDH. We will also show that there is a Baire CDH-subset of the reals that is not Polish. This is joint work with Michael Hru\v{s}\'ak and Rodrigo Hernandez-Guti\'errez.

countable dense homogeneous, $\lambda$-set, real line