our assumptions about $X$.

To build the $K_s$: For $K_{(\,)}$, just use the assumption that $X$ is not totally disconnected. Now, say we are given $K_s$. Choose $x_0,x_1\in K_s$ with $x_0 \ne x_1$. Then find disjoint relatively open $U_0,U_1 \subseteq K_s$ with each $x_\ell \in U_\ell$ and ${\mathrm{diam}}(\pi(U_\ell)) \le 2^{- {\mathrm{lh}}(s) - 1}$. Then find relatively open $V_\ell  \subseteq K_s$ with $x_\ell \in V_\ell \subseteq  \overline{V_\ell}\subseteq U_\ell$. Then, let $K_{s{^{\mathord{\frown}}}\ell}$ be the connected component of the point $x_\ell$ in the space $\overline{V_\ell}$, and note that $K_{s{^{\mathord{\frown}}}\ell}$ cannot be a singleton.

The CSWP: Two Reductions {#sec-cswp-reduce}
========================

These reductions were described in [@KU]: Using the standard theory of function algebras (see [@GAM; @GA]), we can reduce the CSWP to the study of idempotents, and we can reduce the study of the CSWP in LOTSes to the separable case.

If $f \in C(X)$, then $f$ is an *idempotent* iff $f^2 = f$; equivalently iff $f$ is the characteristic function of some clopen set. An idempotent is called *nontrivial* iff it is not the identically $0$ or the identically $1$ function. As with other proofs of the CSWP [@HK1; @KU], we shall proceed by considering idempotents. Following [@KU],

The compact space $X$ has 