Forcing Axioms and the Continuum Hypothesis

One way to formulate the Baire Category Theorem is that no compact space can be covered by countably many nowhere dense sets. Soon after Cohen’s discovery of forcing, it was realized that it was natural to consider strengthenings of this statement in which one replaces countably many with א1-many. Even taking the compact space to be the unit interval, this already implies the failure of the Continuum Hypothesis and therefore is a statement not provable in ZFC. Additionally, there are ZFC examples of compact spaces which can be covered by א1 many nowhere dense sets. For instance if K is the one point compactification of an uncountable discrete set, then K can be covered by א1 many nowhere dense sets. Hence some restriction must be placed on the class of compact spaces in order to obtain even a consistent statement. Still, there are natural classes of compact spaces for which the corresponding statement about Baire Category — commonly known as a forcing axiom — is consistent. The first and best known example is Martin’s Axiom for א1 dense sets (MAא1) whose consistency was isolated from solution of Souslin’s problem [19]. This is the forcing axiom for compact spaces which do not contain uncountable families of pairwise disjoint open sets. For broader classes of spaces, it is much more natural to formulate the class and state the corresponding forcing axiom in terms of the equivalent language of forcing notions.