A compactness property for Borel sets

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.

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 $\bigcap_{\alpha<\omega_1}B_{\alpha}=\emptyset$.

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