Some results on gaps

Abstract

In his article in “Open Problems in Topology” Nyikos asked if it is consistent with ▪ = ω 2 that every ⊂ ∗-increasing ω 1-sequence of subsets of ω is the bottom half of some tight (ω 1, ω 2)-gap. He also asked if this can be generalized to higher cardinals and if in addition it is also possible to get ▪ < ▪. Another question from the same article is whether ▪ = ω 1 implies that there is a tight (ω 1, ω 1)-gap in ωω. Positive answers to the second and third questions are presented here. Theorem 2.12 answers the second question from which the answer to the first question follows as a special case. I also prove Theorems 3.3 and 3.4 which can also be seen as refinements of Hausdorff's theorem on (δ 1, δ 1)-gaps when other assumptions are made in addition to ZFC. Regarding the first question to Nyikos, M. Rabus independently constructed a model of ZFC in which ▪ = ▪ = ω 2 and every ⊂ ∗-increasing ω 1-sequence of subsets of ω is the bottom half of some tight (ω 1, ω 2)-gap. But he has not answered the generalized version of the question. Preserving tight gaps through iteration is more difficult than just preserving gaps, and actually involves preserving a condition stronger than tightness, which in turn implies tightness. Using a completely different method from Rabus, I prove Theorem 2.12 answering in full the most general form of Nyikos' second question and also show that it is possible to get MA to hold below a certain cardinal and at the same time preserve tightness in gaps.