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.