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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

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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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9939-1994-1209100-7
PII: S 0002-9939(1994)1209100-7
Keywords: Normal space, independent set, Bohr compactification, totally bounded topology
Article copyright: © Copyright 1994 American Mathematical Society