Set Theory and Topology seminar (BGU)

Speaker: Jan van Mill (UvA).

Title: Nonhomogeneity of remainders.

Abstract: A space $X$ is homogeneous if for any two points $x,y\in X$ there is a homeomorphism $h$ from $X$ onto itself such that $h(x) = y$.

In 1956, Walter Rudin proved that the Čech-Stone remainder $\beta\omega \setminus \omega$, where $\omega$ is the discrete space of positive integers, is not homogeneous under CH. This result was later generalized considerably by Frolik who showed in ZFC that $\beta X\setminus X$ is not homogeneous, for any nonspeudocompact space $X$. Van Douwen and Kunen proved many results that are in the same spirit.

Hence the study of (non)homogeneity of Čech-Stone remainders has a long history. In this talk we are interested in homogeneity properties of arbitrary remainders of topological spaces. We address the following general problem: when does a space have a homogeneous remainder? If $X$ is locally compact, then the Alexandroff 1-point compactification $\alpha X$ of $X$ has a homogeneous remainder. Hence for locally compact spaces, our question has an obvious answer. If $X$ is not locally compact, however, then it need not have a homogeneous remainder, as the topological sum of the space of rational numbers and the space of irrational numbers shows. Hence we consider questions of the following type: if $X$ is homogeneous, and not locally compact, does $X$ have a homogeneous remainder? We will show that if $X$ is countable and nowhere locally compact, then any remainder of $X$ has at most $\mathfrak{c}$ homeomorphisms, where $\mathfrak{c}$ denotes the cardinality of the continuum. From this we get an example of a countable topological group $G$ no remainder of which is homogeneous. We also get new and very simple proofs that familiar Čech-Stone remainders are not homogeneous.

This is joint work with A. V. Arhangel’skii.