The components are the transfinitely iterated stationary quasicomponents

Question

I read that if we have a topological space $(X,\tau)$, $x\in X$, if we define $C$ as the component of $x$ and $Q$ as the quasicomponents of $x$, we have the well known relations $C\subseteq Q$, so I can consider $Q=Q_1$ with the relative topology and consider $Q_2$ that is the quasicomponent of $x$ in $Q_1$, so $C\subseteq Q_2$, more in general I can define recursively the sets $Q_\alpha$ in this way: $\forall\alpha: Q_{\alpha+1}$ is the quasicomponent of $x$ in $Q_\alpha$, and for the limits ordinals $Q_\alpha=\bigcap_{\beta<\alpha} Q_\beta$. At this point we have $C\subseteq Q_\alpha,\forall\alpha$ and there exist a $\beta$ such that $Q_\beta=Q_{\beta+1}$ for matters of cardinality and, miracuously, we have $C=Q_\beta$. I understood the construction, but i have no idea of why must holds $C=Q_\beta$, can you explain it to me or al least provide me with some reference?

Answer 1

If $Q_\beta$ were disconnected, it would have a nontrivial clopen set and thus would have more than one quasicomponent, so $Q_{\beta+1}$ would be smaller than $Q_\beta$. Since $Q_\beta=Q_{\beta+1}$, that means that $Q_\beta$ is connected.

So $Q_\beta$ is a connected set containing $x$, which means $Q_\beta\subseteq C$. We also have $C\subseteq Q_\alpha$ for all $\alpha$ and in particular for $\alpha=\beta$. Thus $Q_\beta=C$.