Free subsemigroups in topological semigroups (Q2655462)

The notion of almost free was introduced for topological groups and later for topological semigroups. \textit{P. M. Gartside} and \textit{R. W. Knight} [Bull. Lond. Math. Soc. 35, No.~5, 624--634 (2003; Zbl 1045.22021)] gave general conditions under which a topological group is almost free. In this paper, the author considers the conditions for topological semigroups. A subset of a topological space is called meagre if it is a countable union of nowhere dense subsets, and the complement of a meagre set is called co-meagre. Let \(S\) be a topological semigroup. For each \(n\in{\mathbb N}\) set \(C_n=\{(s_1,\dots, s_n)\in S^n:\{s_1,\dots,s_n\}\) freely generates a free subsemigroup of \(S\}\). \(S\) is said to be almost free for each \(n\geq2\) if the set \(C_n\) is not meagre and is co-meagre in \(S^n\). He proves for a complete metrizable topological semigroup \(S\): if \(S\) contains a free dense subsemigroup and any neighborhood of the identity is non-commutative, then \(S\) is almost free. As applications, he shows that the semigroup of all transformations of countable degree under usual composition and the semigroup of all automaton transformations over a finite alphabet are almost free.