For compact Hausdorff spaces, total disconnectedness and total separatedness are equivalent

Assume first that $X$ is totally separated. Let $C\subseteq X$ be connected and choose distinct points $x,y\in C$ if possible. By [L2] there is a clopen set $U$ with $x\in U$ and $y\mathrm{\notin }U$. Then $C\cap U$ and $C\setminus U$ are disjoint nonempty clopen subsets of $C$, contradicting connectedness. Hence every connected component is a singleton, so $X$ is totally disconnected.

Assume now that $X$ is totally disconnected. By [L1], quasicomponents also are singletons. For distinct points $x\mathrm{\ne }y$, the definition of quasicomponent therefore supplies a clopen set containing $x$ but not $y$. This is exactly total separatedness by [L2].

Steps 1.1 and 1.2 prove the equivalence for compact Hausdorff spaces.