prove in Section \[sec-cswp\]:

\[thm-three-lots\] If $X$ is compact and $X \subseteq L_0 \times L_1 \times L_2$, where $L_0,L_1,L_2$ are LOTSes, then $X$ has the CSWP iff $X$ does not contain a copy of the Cantor set.

Here, we may assume that $L_0,L_1,L_2$ are compact (otherwise, replace them by the projections of $X$). It is unknown whether the product of two spaces with the CSWP must also have the CSWP. Even if this turns out to be true, Theorem \[thm-three-lots\] is not immediate from Theorem \[thm-lots\], since $X$ is an arbitrary compact subset of the product, and $L_0,L_1,L_2$ may fail the CSWP (i.e., have Cantor subsets).

By a slightly different argument, we shall show in Section \[sec-powers\]:

\[thm-das-power\] If $L$ is the double arrow space, then $L^n$ has the CSWP for every finite $n$.

Theorems \[thm-three-lots\] and \[thm-das-power\] are proved using some results from Section \[sec-tight\] about spaces which have scattered-to-one maps onto metric spaces. In Theorem \[thm-das-power\], there is a natural $f : L^n {\twoheadrightarrow}[0,1]^n$ for which the inverse of each point is scattered (and of size $2^n$). In Theorem \[thm-three-lots\], the $L_j$ need not have any scattered-to-one maps onto metric spaces, but a standard argument using measures reduces 