${{\mathfrak K}}$ is *local* iff ${{\mathfrak K}}$ is closed-hereditary *and* for every compact $X$: if $X$ is covered by open sets whose closures lie in ${{\mathfrak K}}$, then $X \in {{\mathfrak K}}$.

Classes of compacta which restrict cardinal functions (first countable, second countable, countable tightness, etc.) are clearly local, whereas the class of compacta which are homeomorphic to a LOTS is closed-hereditary, but not local.

It is easily seen that the CSWP is closed-hereditary; this is Lemma 1.3 of [@KU], but the proof is implicit in Rudin [@RUD1]. Thus, to prove part (1) of Theorem \[thm-rudin\] in [@RUD1], it was sufficient to show that the Cantor set itself fails the CSWP.

The removable spaces form a local class (see Section \[sec-remov\]). It is unknown whether the CSWP is a local property. A proof that it is local cannot be completely trivial. For example, locality would imply that the failure of the CSWP for ${{\mathbb T}}$ yields the failure of the CSWP for an arc $A \subseteq {{\mathbb T}}$. Now, $A$ does in fact fail the CSWP, since it contains a Cantor set, but we do not know how to construct a counter-example on $A$ directly from the polynomial algebra ${{\mathcal 