not scattered, so as Rudin pointed out:

\[cor-metric\] If $X$ is compact metric, then $X$ satisfies the CSWP iff $X$ does not contain a copy of the Cantor set.

One might conjecture that this corollary holds for all compact $X$, but that was refuted in 1960 by Hoffman and Singer [@HS] (see also [@GA; @HO]); their results imply that any compactum containing $\beta{{\mathbb N}}$ fails the CSWP.

However, the corollary does hold for some more “reasonable” classes of spaces. Kunen [@KU] showed in 2004:

\[thm-lots\] If $X$ is a compact LOTS, then $X$ satisfies the CSWP iff $X$ does not contain a copy of the Cantor set.

As usual, a LOTS is a linearly ordered topological space. Of course, the $\rightarrow$ of this result is clear from Theorem \[thm-rudin\]; only the $\leftarrow$ was new. This theorem shows that there are some non-scattered spaces with the CSWP, such as the double arrow space of Alexandroff and Urysohn (see Definition \[def-das\], or [@AU], p. 76).

One can now ask whether there are further classes of “*reasonable*” spaces for which results such as Corollary \[cor-metric\] and Theorem \[thm-lots\] hold. We do not know the best possible result along this line, but we shall 