Every uncountable abelian group admits a nonnormal group topology

Trigos-Arrieta, F. Javier

doi:10.1090/S0002-9939-1994-1209100-7

Every uncountable abelian group admits a nonnormal group topology
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by F. Javier Trigos-Arrieta PDF

Proc. Amer. Math. Soc. 122 (1994), 907-909 Request permission

Abstract:

If G is a locally compact Abelian group, let ${{\mathbf {G}}^ + }$ denote the underlying group of G equipped with the weakest topology that makes all the continuous characters of G continuous. Thus defined, ${{\mathbf {G}}^ + }$ is a totally bounded topological group. We prove: Theorem. ${{\mathbf {G}}^ + }$ is normal if and only if G is $\sigma$-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.

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