Closed subgroups of the infinite symmetric group

Let S = Sym(Ω) be the group of all permutations of a countably infinite set Ω, and for subgroups G ₁, G ₂ ≤ S let us write G ₁ ≈ G ₂ if there exists a finite set U ⊆ S such that 〈G₁ ∪ U〉 = 〈G₂ ∪ U〉. It is shown 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 statements about pointwise stabilizer subgroups G_(Γ) of finite subsets Γ ⊆ Ω holds: (i) For every finite set Γ, the subgroup G_(Γ) has at least one infinite orbit in Ω (ii) There exist finite sets Γ such that all orbits of G_(Γ) are finite, but none such that the cardinalities of these orbits have a common finite bound. (iii) There exist finite sets Γ such that the cardinalities of the orbits of G_(Γ) have a common finite bound, but none such that G_(Γ) = {1}. (iv) There exist finite sets Γ such that G_(Γ) = {1}. Some related results and topics for further investigation are noted.