Generating the infinite symmetric group using a closed subgroup and the least number of other elements

Abstract:

Let $S_{\infty }$ denote the symmetric group on the natural numbers $\mathbb {N}$. Then $S_{\infty }$ is a Polish group with the topology inherited from $\mathbb {N}^{\mathbb {N}}$ with the product topology and the discrete topology on $\mathbb {N}$. Let $\mathfrak {d}$ denote the least cardinality of a dominating family for $\mathbb {N}^{\mathbb {N}}$ and let $\mathfrak {c}$ denote the continuum. Using theorems of Galvin, and Bergman and Shelah we prove that if $G$ is any subgroup of $S_{\infty }$ that is closed in the above topology and $H$ is a subset of $S_{\infty }$ with least cardinality such that $G\cup H$ generates $S_{\infty }$, then $|H|\in \{0,1,\mathfrak {d},\mathfrak {c}\}$.