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Every uncountable abelian group admits a nonnormal group topology

Author: F. Javier Trigos-Arrieta
Journal: Proc. Amer. Math. Soc. 122 (1994), 907-909
MSC: Primary 22B05; Secondary 54A10, 54A25, 54H11
MathSciNet review: 1209100
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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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Keywords: Normal space, independent set, Bohr compactification, totally bounded topology
Article copyright: © Copyright 1994 American Mathematical Society

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