This is an update on, and expansion of, our paper Open problems on βω in the book Open Problems in Topology.

We study the existence of universal autohomeomorphisms of N ^*. We prove that the Continuum Hypothesis (CH) implies there is such an auto-homeomorphism and show that there are none in any model where all auto-homeomorphisms of N ^* are trivial.

The classical Erdös spaces are obtained as the subspaces of real sep- arable Hilbert space consisting of the points with all coordinates rational or all coordinates irrational, respectively. One can create variations by specifying in which set each coordinate is allowed to vary. We investigate the homogeneity of the resulting subspaces. Our two main results are: in case all coordinates are allowed to vary in the same set the subspace need not be homogeneous, and by specifying different sets for different coordinates it is possible to create a rigid subspace.