Applications of cohomology to questions in set theory i: hausdorff gaps

Abstract

We explore an application of homological algebra to set theoretic objects by developing a cohomology theory for Hausdorff gaps. The cohomology theory is introduced with enough generality to be applicable to other questions in set theory. For gaps, this leads to a natural equivalence notion about which we answer questions by constructing many simultaneous gaps. The first result is proved by constructing countably many gaps such that the union of any combination corresponding to a subset of $\omega$ is again a gap. This is done in ZFC. New combinatorial hypotheses related to club ($\clubsuit$) are introduced to prove the second result which increases the number of gaps to $\aleph_1$. The presentation of these results does not depend on the development of the cohomology theory. Additionally, the notion of an incollapsible gap is introduced and the existence of such a gap is shown to be independent of ZFC.