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Normality and the Weak Topology on a Locally Compact Abelian Group
Author(s) -
TRIGOSARRIETA F. JAVIER
Publication year - 1994
Publication title -
annals of the new york academy of sciences
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.712
H-Index - 248
eISSN - 1749-6632
pISSN - 0077-8923
DOI - 10.1111/j.1749-6632.1994.tb44154.x
Subject(s) - locally compact space , locally compact group , mathematics , abelian group , embedding , second countable space , group (periodic table) , topological group , bounded function , topology (electrical circuits) , discrete mathematics , compact group , discrete group , combinatorics , pure mathematics , lie group , computer science , mathematical analysis , physics , artificial intelligence , quantum mechanics
If G is a locally compact Abelian group, let G + denote the underlying group of G equipped with the weakest topology that makes all the continuous characters of G continuous. Thus defined, G + is a totally bounded topological group. We present the original proof of the following result: T heorem : For G discrete, G + is normal if and only if G is countable. C orollary : For G locally compact, G + is normal if and only if G is s̀‐compact. Our technics make use of the notion of z ‐embedding. The 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.