Maybe I'm missing something, but this seems like a mistake.
Exercise 2.9 (page 36) says:
Show that an open cover $\{U_\alpha\}_{\alpha\in A}$ is locally finite if and only if each $U_\alpha$ intersects $U_\beta$ for only finitely many $\beta$.
And he defines locally finite as: A collection of subsets $\{U_\alpha\}_{\alpha\in A}$ of a topological space $X$ such that each point $p\in X$ has a neighborhood that intersects at most finitely many of the sets $U_\alpha$.
But what about the reals $R$ with the cover $U_0=R$, $U_i = (i, i+1)$ for $i=1,2,...$? Every point has a neighborhood that intersects at most 2 sets so it's locally finite, yet the set $U_0=R$ intersects every other $U_i$.
What am I missing?
There is no such exercise on page 36, in either the first or second edition of my book. You must be using one of the pirated draft versions of the first edition, which somebody posted illegally on the internet. Those are full of mistakes and come with no guarantees.