#### RIS ID

6922

#### 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

*A*

_{2}(

*X*) is locally compact if an only if

*F*

_{2}(

*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

*F*

_{4}(

*X*) is locally compact if and only if

*F*(

_{n}*X*) has pointwise countable type for each

*n*ω if and only if

*F*

_{4}(

*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

*A*

_{2}(

*X*) has pointwise countable type if and only if

*F*

_{2}(

*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

*F*

_{2}(

*X*) is locally compact if and only if

*F*

_{3}(

*X*) is locally compact, and that

*F*

_{2}(

*X*) has pointwise countable type if and only if

*F*

_{3}(

*X*) has pointwise countable type.

## Publication Details

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