The whole question is like this:
1-4. Let M be a topological manifold, and let U be an open cover of M .
(a) Assuming that each set in U intersects only finitely many others, show that U is locally finite.
(b) Give an example to show that the converse to (a) may be false.
(c) Now assume that the sets in U are pre compact in M; and prove the con- verse: if U is locally finite, then each set in U intersects only finitely many others.
I'm still tackling the part b).
From John Lee, "Smooth manifold" Problem 1-4