Every uncountable abelian group admits a nonnormal group topology

Abstract

If G is a locally compact Abelian group, let G + {{\mathbf {G}}^ + } denote the underlying group of G equipped with the weakest topology that makes all the continuous characters of G continuous. Thus defined, G + {{\mathbf {G}}^ + } is a totally bounded topological group. We prove: Theorem. G + {{\mathbf {G}}^ + } is normal if and only if G is σ \sigma -compact. When G is discrete, this theorem answers in the negative a question posed in 1990 by E. van Douwen, and it partially solves a problem posed in 1945 by A. Markov.