A compact group which is not Valdivia compact

Abstract

A compact space $K$ is {\em Valdivia compact} if it can be embedded in a Tikhonov cube $I^A$ in such a way that the intersection $K\cap\Sigma$ is dense in $K$, where $\Sigma$ is the sigma-product (= the set of points with countably many non-zero coordinates). We show that there exists a compact connected Abelian group of weight $\o_1$ which is not Valdivia compact, and deduce that Valdivia compact spaces are not preserved by open maps.