the proof of Theorem \[thm-three-lots\] to the case where the $L_j$ are separable (see Section \[sec-cswp-reduce\]), in which case $X$ must have an eight-to-one map onto a compact metric space.

If $L_0,L_1,L_2$ are separable in Theorem \[thm-three-lots\], then $X$ must also be first countable, and hence “small” in the cardinal functions sense (see Juhász [@JU]). However, we do not believe that there is a notion of “reasonable” involving only cardinal functions. In [@HK2] it is shown that in some models of set theory, there is a compact $X$ which does not contain Cantor subsets and which fails the CSWP, such that $X$ is both hereditarily separable and hereditarily Lindelöf (and hence also first countable). In these models, $2^{\aleph_0} = \aleph_1$ and the standard cardinal functions of our $X$ (all either $\aleph_0$ or $\aleph_1$) are the least possible among non-metric compacta.

Section \[sec-LOTS\] reviews some elementary fact about LOTSes. Section \[sec-remov\] discusses the notion of a *removable space* defined in [@HK1]; this is a strengthening of the CSWP used in Section \[sec-powers\].

\[def-local\] Let ${{\mathfrak K}}$ be a class of compact spaces. ${{\mathfrak K}}$ is *closed-hereditary* iff every closed subspace of a space in ${{\mathfrak K}}$ is also in ${{\mathfrak K}}$. 