Local compactness in free topological groups

Authors

Peter Nickolas, University of WollongongFollow
Mikhail Tkachenko, Universidad Autonoma Metropolitana - Iztapalapa

RIS ID

6922

Publication Details

Nickolas, P. & Tkachenko, M. (2003). Local compactness in free topological groups. Bulletin of the Australian Mathematical Society, 68 (2), 243-265.

Abstract

We show that the subspace A_n(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 F_n(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 F_n(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 A_n(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.