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Local compactness in free topological groups

Published online by Cambridge University Press:  17 April 2009

Peter Nickolas  and
Mikhail Tkachenko
Peter Nickolas
Affiliation:
School of Mathematics and Applied Statistics, University of Wollongong, New South Wales 2522, Australia, e-mail: [email protected]
Mikhail Tkachenko
Affiliation:
Departamento de Matemáticas, Universidad Autónoma Metropolitana, Av. San Rafael Atlixco # 186, Col. Vicentina, C.P. 09340, Iztapalapa, Mexico, D.F., e-mail: [email protected]
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Abstract

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We show that the subspace An(X) of the free Abelian topological group A(X) on a Tychonoff space X is locally compact for each n ∈ ω if and only if A2(X) is locally compact if an only if F2(X) is locally compact if and only if X is the topological sum of a compact space and a discrete space. It is also proved that the subspace Fn(X) of the free topological group F(X) is locally compact for each n ∈ ω if and only if F4(X) is locally compact if and only if Fn(X) has pointwise countable type for each n ∈ ω if and only if F4(X) has pointwise countable type if and only if X is either compact or discrete, thus refining a result by Pestov and Yamada. We further show that An(X) has pointwise countable type for each n ∈ ω if and only if A2(X) has pointwise countable type if and only if F2(X) has pointwise countable type if and only if there exists a compact set C of countable character in X such that the complement X \ C is discrete. Finally, we show that F2(X) is locally compact if and only if F3(X) is locally compact, and that F2(X) has pointwise countable type if and only if F3(X) has pointwise countable type.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 68 , Issue 2 , October 2003 , pp. 243 - 265
Copyright
Copyright © Australian Mathematical Society 2003

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