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 <G
1 ∪ U> = <G
2 ∪ 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.