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

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

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

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