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Let denote the symmetric group on the natural numbers . Then is a Polish group with the topology inherited from with the product topology and the discrete topology on . Let denote the least cardinality of a dominating family for and let denote the continuum. Using theorems of Galvin, and Bergman and Shelah we prove that if is any subgroup of that is closed in the above topology and is a subset of with least cardinality such that generates , then .

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