Comment.Math.Univ.Carolin. 39,1 (1998)185–195 185 Continuous functions between Isbell-Mrówka spaces

Abstract. Let Ψ(Σ) be the Isbell-Mrówka space associated to the MAD-family Σ. We show that if G is a countable subgroup of the group S(ω) of all permutations of ω, then there is a MAD-family Σ such that every f ∈ G can be extended to an autohomeo-morphism of Ψ(Σ). For a MAD-family Σ, we set Inv(Σ) = {f ∈ S(ω) : f [A] ∈ Σ for all A ∈ Σ}. It is shown that for every f ∈ S(ω) there is a MAD-family Σ such that f ∈ Inv(Σ). As a consequence of this result we have that there is a MAD-family Σ such that n+ A ∈ Σ whenever A ∈ Σ and n < ω, where n+ A = {n+ a: a ∈ A} for n < ω. We also notice that there is no MAD-family Σ such that n ·A ∈ Σ whenever A ∈ Σ and 1 ≤ n < ω, where n · A = {n · a: a ∈ A} for 1 ≤ n < ω. Several open questions are listed