Closed subgroups of the infinite symmetric group.
If \(\Omega\) is an infinite set and we regard it as a discrete topological space, then the set \(\Omega^\Omega\) of all functions \(\Omega\to\Omega\) becomes a topological space under the function topology (in this topology, a subbasis of open sets is given by the sets \(\{f\in\Omega^\Omega:f(\alpha)=\beta\), \(\alpha,\beta\in\Omega\}\) and the closure of a set \(U\subset\Omega^\Omega\) consists of all maps \(f\) such that, for every finite subset \(\Gamma\subseteq\Omega\), there exists an element of \(U\) agreeing with \(f\) at all members of \(\Gamma\)). Let now \(\Omega\) be a countably infinite set and \(S=\text{Sym}(\Omega)\) the group of all permutations of \(\Omega\). For \(G_1,G_2\leq S\) let us write \(G_1\approx G_2\) if there exists a finite set \(F\subseteq G\) such that \(\langle G_1\cup F\rangle=\langle G_2\cup F\rangle\). Let \(\lambda\) be the least cardinal such that for some finite subset \(\Gamma\subseteq\Omega\), all orbits of \(G_{(\Gamma)}\) (pointwise stabilizer subgroup of \(\Gamma\)), in \(\Omega\) have cardinality \(<\lambda\). In this paper the authors show that the subgroups closed in the function topology on \(S\) lie in precisely four equivalence classes under this relation. -- Which of these classes a closed subgroup \(G\) belongs to depends on which of the following conditions on \(\lambda\) holds: (i) \(\lambda=\aleph_1\), (ii) \(\lambda=\aleph_0\), (iii) \(3\leq\lambda<\aleph_0\), and (iv) \(\lambda=2\).
full permutation group on countably infinite set
cardinalities of orbits of stabilizers of finite sets
scientific article