Let $N$ be an infinite subset of $Q$ that is closed in $\bigcup_{\beta<\alpha}X_\beta$; then every point of $\left(\bigcup_{\beta<\alpha}X_\beta\right)\setminus N$ has a nbhd disjoint from $N$. In particular, for each $\beta<\alpha$ there is a $B_\beta\in\mathcal{B}_{x_\beta}$ such that $N\cap B_\beta=\varnothing$. If you go on to read the construction of the local base $\mathcal{B}_{x_\alpha}$, you’ll see that each $\mathcal{B}_{x_\beta}$ is countable and nested, so that if $N\cap B_\beta=\varnothing$, then $N$ intersects at most finitely many members of $\mathcal{B}_{x_\beta}$. By definition, therefore, $N\notin\mathcal{N}_\beta$ for any $\beta<\alpha$, i.e., $N\in\mathcal{N}\setminus\bigcup_{\beta<\alpha}\mathcal{N}_\beta$.
(By the way, Raushan Buzyakova is a woman.)