Nonhomogeneity of Remainders, Ii

We present an example of a separable metrizable topological group G having the property that no remainder of it is (topologically) homogeneous. 1. Introduction. All topological spaces under discussion are Tychonoff. A space X is homogeneous if for any two points x, y ∈ X there is a homeomorphism h from X onto itself such that h(x) = y. If bX is a com-pactification of a space X, then bX \ X is called its remainder. In 1956, Walter Rudin [13] proved that the Čech–Stone remainder βω \ω, where ω is the discrete space of non-negative integers, is not homogeneous under CH. This result was later generalized considerably by Frolík [9] who showed in ZFC that βX \ X is not homogeneous, for any nonpseudocompact space X. For other results in the same spirit, see e.g. [6], [7], [10]. Hence the study of (non)homogeneity of Čech–Stone remainders has a long history. In this note we continue our study begun in [4] concerning the (non)homogeneity of arbitrary remainders of topological spaces. Special attention is given to remainders of non-locally compact topological groups. For some recent facts on such remainders, see Arhangel'skii [1] and [2]. One of them, established in [1], is: every remainder of a topological group is either Lindelöf or pseudocompact. The aim of this note is to present an example of a separable metrizable topological group G no remainder of which is homogeneous. The first examples of topological groups that share this property can be found in [4]; these examples have various interesting properties but are not metrizable.