Introduction. For f ∈ ωω and an ultrafilter U on ω define f(U) = {X ⊆ ω: f−1(X) ∈ U}, and for f, g ∈ ωω we say that f = g mod U if there is X ∈ U such that f(n) = g(n) for n ∈ X.

We say that U is Hausdorff if for any two functions f, g ∈ ωω, if f(U) = g(U) then f = g mod U.

Let FtO be the collection of all finite-to-one functions f ∈ ωω. Recall that an ultrafilter U is a p-point if for every function f ∈ ωω either there is n such that f−1({n}) ∈ U or there exists g ∈ FtO such that f = g mod U. Similarly, U is Ramsey if for every function f ∈ ωω either there is n such that f−1({n}) ∈ U or there exists a one-to-one function g ∈ ωω such that f = g mod U.

In this paper we will assume that all ultrafilters U, and their images f(U) are non-principal.

It is worth mentioning that the following appears as an exercise in [7]. If f(U) = U then f = id mod U. Therefore, if U is not Hausdorff, then this is witnessed by two functions, both not one-to-one mod U. It follows from it that Ramsey ultrafilters are Hausdorff.