On page 84 of Adam's book on Sobolev spaces he states the 'uniform $C^m$-regularity condition' for a domain $\Omega$. This condition states that when we have a locally finite open cover $\{U_j\}$ of the boundary of $\Omega$, then for some finite $R$, every collection of $R+1$ of the sets $U_j$ has empty intersection.
But if every collection of some $R+1$ of these sets has empty intersection, it seems they cannot cover the boundary! I.e. if this collection is indeed going to cover the boundary then each open set $U_j$ will have to intersect with at least one other set $U_i$, $i \neq j$.
How am I misinterpreting this statement?
The condition says that every subcollection of $\{U_{j_k}\}_{k=1}^{R+1} \subset \{U_j\}$ has empty intersection, so we require $$ \bigcap_{k=1}^{R+1} U_{j_k} = \emptyset. $$ This is not the same as requiring pairwise empty intersection, which would assert that $U_{j_k} \cap U_{j_{\ell}} = \emptyset$ for all $k \neq \ell.$
To get a better understanding of what this means, observe it is equivalent to requiring to finding a locally finite covering where each point is contained in at most $R$ of the open sets. So if $\{U_i\}$ is the given cover, for each $x \in \partial \Omega$ we have, $$ \sum_j \chi_{U_j}(x) \leq R. $$ In particular if the cover itself is finite, taking $R$ to be the number of open sets of the cover we get this property is satisfied.