Strong version of Baire Category Theorem

Question

We know that in a complete metric space or compact Hausdorff space the intersection of $\omega$-many open dense sets is dense.

In such spaces is the intersection of fewer than $2^\omega$-many open dense sets dense? What about in compact metric spaces?

Answer 1

The statement that for $X$ compact Hausdorff that is ccc (there is no uncountable family of non-empty pairwise disjoint open sets) every intersection of $<2^{\aleph_0}$ open dense sets is dense is equivalent to Martin's axiom MA.

This is independent of the usual ZFC axioms of set theory, and is a very well-studied axiom (especially its consequences in topology and measure theory).

Without the ccc condition on $X$, the statement is false in ZFC. For compact metrisable spaces ccc follows (as these are separable, and this implies ccc), so it would be a consequence of MA. I don't think it is a ZFC fact, though I don't have a consistent counterexample at hand.

Answer 2

It is perhaps also worth noting that the statement 'for all nonempty separable compact Hausdorff spaces, the intersection of $<2^{\aleph_0}$ open dense sets is dense' is equivalent to Martin's Axiom for $\sigma$-centered posets. A subset $C$ of a poset $P$ is centered if for any $p_1,\ldots,p_n\in C$ there is a $q\in P$ such that $q\leq p_i$ for $i\leq n$. A poset is $\sigma$-centered if it is the union of countably many centered sets.

As with MA, this statement is independent of ZFC and has many applications to topology and analysis. It is obviously implied by MA, but is not equivalent to it.