A compactness property for Borel sets

Question

Is the following generalised compactness property of Borel sets in a Polish space consistent with ZFC?

($*$) Let $\mathcal{B}$ be a family of $\aleph_1$-many Borel sets. If $\bigcap \mathcal{B} = \emptyset$, then, for some countable $\mathcal{C} \subseteq \mathcal{B}$, it holds that $\bigcap \mathcal{C} = \emptyset$.

In the special case that $\mathcal{B}$ is a family of open sets, it is not hard to show that property ($*$) is a consequence of $\neg\mathrm{CH} + \mathrm{MA}$ (where $\mathrm{CH}$ is the Continuum Hypothesis and $\mathrm{MA}$ is Martin's Axiom)—more specifically, a consequence of $\mathrm{MA}_{\aleph_1}$. Also, trivially, this special case of ($*$) itself implies $\neg \mathrm{CH}$.

However, I am stuck as to the consistency of ($*$) for general Borel sets. Actually, I have been unable to show consistency even when $\mathcal{B}$ is restricted to families of $F_\sigma$ sets.

Answer 1

It is worth noting that a construction provided by Hausdorff more than a hundred years before the result of Kubis and Vejnar also provides a counterexample. A Hausdorff gap is a family of subsets of the natural numbers $A_\xi, B_\xi$ for $\xi\in\omega_1$ such that $A_\xi \subseteq^* A_\eta \subseteq^* B_\eta \subseteq^* B_\xi$ for $\xi< \eta$ but such that there is no subset of $\mathbb N$ such that $ A_\xi \subseteq^* X \subseteq^* B_\xi$ for all $\xi$. (Here $\subseteq^*$ means inclusion except for a finite set.) Letting $S_\xi$ be the Borel set of all $X\subseteq \mathbb N$ such that $ A_\xi \subseteq^* X \subseteq^* B_\xi$ yields the counterexample.

Answer 2

Here is a even simpler example.

Let $\{x_{\alpha} \}_{\alpha\in \omega_1}$ be an increasing chain in the Turing degrees. For every $\alpha$, let $B_{\alpha}=\{y\mid y\geq_T x_{\alpha}\}$. Each $B_{\alpha}$ is a boldface $\Sigma^0_3$ set.

Then for any countable ordinal $\beta$, $\bigcap_{\alpha<\beta}B_{\alpha}$ is not empty but $\bigcap_{\alpha<\omega_1}B_{\alpha}=\emptyset$.

Answer 3

This compactness property is never true, even for collections of $F_\sigma$ subsets of an uncountable Polish space. One way to see this is to fix your favorite example of an $F_\sigma$ graph $G$ with clique number $\aleph_1$ and a maximal $G$-clique $K$. Then let your family be $\{G_x : x \in K\}$, where $G_x$ is the set of neighbors of $x$ in $G$.

For a simple example of such a graph, see, e.g., http://www.math.cas.cz/preprint/pre-207.pdf