on $X$ such that ${\mathrm{supt}}(\mu)$ fails the CSWP, let $\mu \pi_\alpha{^{-1}}$ be the induced measure on $L_\alpha$, let $H_\alpha = {\mathrm{supt}}(\mu \pi_\alpha{^{-1}})$, which is separable, and note that ${\mathrm{supt}}(\mu) \subseteq X \cap \prod_{\alpha < \kappa} H_\alpha$.

\[lemma-one-S\] For any $\kappa \le \omega$: Suppose that there is a compact $X \subseteq \prod_{\alpha < \kappa} L_\alpha$, where each $L_\alpha$ is a LOTS, $X$ has no Cantor subset, and $X$ does not have the CSWP. Then there is such an $X$ which is a subset of $(I_S)^\kappa$ for some $S \subseteq (0,1)$.

By Lemma \[lemma-prod-cswp\], we may assume that each $L_\alpha$ is separable and compact. Now, let $L$ be the compact separable LOTS obtained by placing the $L_\alpha$ end-to-end, adding a point $\infty$ in the case that $\kappa  = \omega$. Then we may assume that $X \subseteq L^\kappa$. Finally, replace $L$ by an $I_S$ using Lemma \[lemma-LOTS-standard\](3).

The CSWP and Tightness {#sec-cswp}
======================

We show here how one can use the concepts from Section \[sec-tight\] to produce idempotents, and thus to prove the CSWP.

\[def-phat\] Assume that $\pi: X \to Y$, where $X,Y$ are compact. Then, for $f \in C(X)$, define $\widehat f = (\pi\times f)(X)$; that is, $$\widehat f \ =\  \{(\pi(x), 