2.3pt \hbox{\scriptsize $\mathrm{c}$}}\; }C(X)$ and there is some $h \in {{\mathcal A}}$ such that either $\Re(h(X))$ or $\Im(h(X))$ is not connected. Then ${{\mathcal A}}$ contains a non-trivial idempotent.

This is easy to prove using Runge’s Theorem; see Lemma 2.5 of [@KU], but the method was also used in [@RUD1] and [@HS].

It remains to describe how to obtain such an $h$. If $X$ is scattered, then $\Re(h(X))$ is scattered also, so any $h$ for which $\Re(h(X))$ is not a singleton will do; this is essentially the argument of [@RUD2]. In some other cases, we can obtain $h$ using a tight map of $X$ onto a metric space; this is described in Section \[sec-cswp\].

We now turn to the second reduction. As in §5 of [@KU],

If $\mu$ is a regular complex Borel measure on the compact space $X$, then $|\mu|$ denotes its total variation, and ${\mathrm{supt}}(\mu) = {\mathrm{supt}}(|\mu|)$ denotes its (closed) support; that is, ${\mathrm{supt}}(\mu) = X \setminus \bigcup \{U\subseteq X :\; $U$ \mbox{ is open }
\ \ \& \ \ |\mu|(U) = 0\}$.

Considering measure orthogonal to ${{\mathcal A}}$, we get:

\[lemma-supt-cswp\] Assume that $X$ is compact and that ${\mathrm{supt}}(\mu)$ has the CSWP for all regular Borel measure 