Exercise 13, § 4

Let $X$ be a locally compact space and $U$ an open subset of $X$. Let $B$ be the union of $U$ and the relatively compact components of $CU$. Show that $B$ is open and is the union of $U$ and those subsets of $CU$ which are both open and compact in $CU$. (Embed $X$ in its one-point compactification $X' = X \cup \{ \omega \}$, observe that in $X' - U$ the component of $\omega$ is $\{ \omega \} \cup CB$, and use Proposition 6 of no. 4).