Nonhomogeneity in products with [formula omitted]-spaces

Abstract

This note is a contribution to the study of large compact homogeneous spaces (see [5], [7]) and homogeneity properties of products of compact spaces in particular. It is an application of results from [4] and an attempt to draw more attention to these results. Infinite products of zero-dimensional first-countable spaces are homogeneous ([2]), but the homogeneity of certain spaces cannot be improved by taking powers, or even products. Consider the following property of a space X. (*) X × Y is not homogeneous for any compact space Y . This property is shared by (ω1 +1) ([3]) and the ‘topologist’s sine curve’ (Motorov proved it is not a retract of any compact homogeneous space). In [5] Kunen pointed out that it is not known whether (*) holds for all – or any – infinite F-space(s) and proved a weakening of (*) for infinite F-spaces. His theorem implies, for example, that a product of any family of (nontrivial) F-spaces is never homogeneous. Instead of F-spaces we work with a slightly larger class of compact βN-spaces introduced in [1] (see below for definitions). Kunen’s result and its proof remain correct if ‘F-space’ is replaced by ‘βN-space’ everywhere.