I came across this lemma in Lee's 'Introduction to Smooth Manifolds'. The lemma seems simple enough to prove, but I just can't seem to prove it. It's frustrating me because I know it must be simple.
The lemma is: An open cover $\{U_{\alpha}\}$ of a topological space $X$ is locally finite if and only if each $U_{\alpha}$ intersects $U_{\beta}$ for only finitely many $\beta$.
Lee then asks to Give a counterexample if the sets of the cover are not assumed to be open.
Does anyone have any glaringly obvious things to point out I'm missing?
I have thought about trying to prove the necessary statement via contradiction. Ie suppose the open cover is locally finite but assume there exists a $U_{\alpha}$ such that $U_{\alpha}$ intersects infinitely many $U_{\beta}$-s. The required result would follow easily (I think) if I was able to show that there is a common point $p$ in this infinite intersection' - but I'm not sure if I can conclude this.
What you've stated is not true. Take, for example the topological space $\mathbb R \times (0, 3) \subseteq \mathbb R^2$. The following open cover is locally finite but the first set intersects infinitely many others:
I think what you're referring to is Problem 1-4 on page 31. There you're asked to prove the lemma you've stated under the assumption that the sets in the open cover are precompact.