[FOM] free ultrafilters

Freek Wiedijk freek at cs.ru.nl
Mon Oct 16 09:11:48 EDT 2006

Bill Taylor:

>Is it possible/easy to describe in some way, how to have two
>non-isomorphic free ultrafilters on the integers, where
>isomorphism is defined with respect to permutations on the
>integers.

I think I remember having been told that if you put the
Cech-Stone topology on the collection of free ultrafilters
over the integers, then that topological space -- called
"\beta\omega - \omega" -- is _not_ homogeneous (in the sense
that you can't map any point to any other point using an
automorphism of the space), which is a little bit surprising
as of course the space of the integers that the Cech-Stone
compactification starts from _is_ homogeneous.

Some googling to find a reference for this led me to:

$\beta\omega - \omega$ is not first order homogeneous" by
Eric K. van Douwen and Jan van Mill, Proceedings of the
American Mathematical Society, Vol. 81, No. 3 (Mar., 1981),
pp. 503-504.

The abstract of this "shorter note" is:

"We find a first order property shared by some but not all
points of $\beta\omega - \omega$."

Freek