On some properties of free topological groups. (Russian) Zbl 0568.22001

Tkachenko, M. G.

Mat. Zametki 37, No. 1, 110-118 (1985).

A subset X of a topological group G is called thin in G if for any neighbourhood V of the unit in G there exists a neighbourhood W of the unit in G such that \(x\cdot W\cdot x^{-1}\subseteq V\), for any \(x\in X\). The basic problem of the paper is to characterize these spaces X (complete regular) for which X is thin in Markov’s free topological group \(F_ M(X)\) on X. Let X be not a P-space (all \(G_{\delta}\)-sets are not open in X). Then X is thin in \(F_ M(X)\) iff X is pseudocompact (i.e., all continuous functions on X are bounded). This main result is generalized to the case that X is an arbitrary space and the author gives the characterization of such X for which \(F_ M(X)\) has an invariant basis.

Reviewer: B.Smarda

MSC:

22A05

Structure of general topological groups

Keywords:

topological group; thin; Markov’s free topological group; invariant basis