Corrigendum to “Gruff ultrafilters” (Topology and its Applications (2016) 210 (355–365) (S0166864116301869) (10.1016/j.topol.2016.08.012))

The authors regret that in Section 4 of the paper [2], the proof of Theorem 4.2 contains a mistake, and the authors do not know whether it can be fixed. The problem is that the terms [Formula presented] of the strong pathway [Formula presented], as defined in the paper, will not be closed under binary set-theoretically definable operations; in fact, they will not even be closed under the join operation (the authors are very grateful to Osvaldo Guzmán for pointing this out, as well as for providing a concrete example of the failure of this closure property). In fact, the same problem arises in an analogous argument from the paper [1], where a similar technique was first implemented. Consequently, the question of whether there are P-points in every Random extension of a model of [Formula presented] (an affirmative answer to which [1] had been considered an established result for quite some time) appears to be open at the moment. Since the proof of Theorem 4.3 in [2] depends on Theorem 4.2, the authors regrettably withdraw the claim made in the statement of this Theorem. We do not know if there exist gruff ultrafilters in every model obtained by adding Random reals to a model of [Formula presented]. We do, however, know that there is a model M of [Formula presented] such that there are gruff ultrafilters in any model obtained by adding any number of random reals to M. To see this it suffices to consider [Formula presented], where [Formula presented] and [Formula presented] denotes the forcing notion for adding [Formula presented]-many Cohen reals to V. Let [Formula presented] be the ([Formula presented]-name for the) forcing notion that adds λ-many Random reals to [Formula presented], where λ is some cardinal of [Formula presented] (equivalently, of V, since [Formula presented] is a c.c.c. forcing notion). Then we claim that in the generic extension [Formula presented] there is a strong pathway. To see this, let [Formula presented] be the [Formula presented]-sequence of Cohen reals added by [Formula presented]. For each [Formula presented], consider the model of Set Theory [Formula presented]. If we let [Formula presented] denote the interpretation of the Borel code b in the model of Set Theory W, and we have two such models [Formula presented], recall that a function [Formula presented] is random over W if [Formula presented] for any null [Formula presented]-set coded in W (i.e. such that [Formula presented]). In particular, if G is an [Formula presented]-generic filter over [Formula presented], and [Formula presented] if and only if [Formula presented], then [Formula presented] is random over [Formula presented], and consequently it is also random over [Formula presented] for every [Formula presented]. It is then easy to verify that the sequence [Formula presented] given by [Formula presented] (for limit α we let [Formula presented] in order to make the sequence continuous) will be a strong pathway: First, as [Formula presented] is a generic extension of V, [Formula presented] has all the closure properties required; moreover, [Formula presented] is not dominated by any real from [Formula presented] and hence it is not dominated by any element of [Formula presented] either. On the other hand, every real in [Formula presented] only depends on [Formula presented] and countably many of the [Formula presented], so [Formula presented] The proof of Theorem 4.3 goes through verbatim upon replacing the Random model with the model [Formula presented], by using the strong pathway defined above. Hence we can conclude the existence of gruff ultrafilters in any model that arises from having [Formula presented], adding [Formula presented] Cohen reals, and subsequently adding some uncountable number of Random reals. (Incidentally, with the same reasoning we can conclude that there are P-points in any such model, which is an unpublished old result of Kunen). The authors would like to apologise for any inconvenience caused.