# Strong version of Baire Category Theorem

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?

• In fact, I think I remember seeing something similar under a name strong Baire category theorem or SBCT . Maybe you can find a few additional references by searching for this. – Martin Sleziak Dec 9 '14 at 19:01
• Indeed, e.g. math.toronto.edu/tall/publications/55.pdf mentions that MA for countable posets is equivalent to this property of dense open sets for the reals. – Henno Brandsma Dec 9 '14 at 20:32

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.
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.
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.