in a manifold with an open cover of precompact balls, local finiteness implies each set in the cover intersects only finitely many others.

Question

I tried to prove the result using contradiction, but got stuck: Let $M$ be a topological manifold, with an open cover $\upsilon$ that is locally finite and $\upsilon$ consists of pre compact open balls in $M$, then, let, if possible, there exists an open set $U \in \upsilon$ s.t. it intersects infintely many elements of $\upsilon$, then we get an element in each of these intersections. But i'm not sure how to proceed from here, I'm unable to utilise the property of pre compactness endowed on the elements of $\upsilon$. Please tell me how to proceed from here.

Answer 1

You have to use the local finiteness of the cover $\nu$. Because it is locally finite, around each point $x$ in $U$, we can find an open neighbourhood, say $V_{x}$ such that only finitely many sets from $\nu$ intersect it.

Now consider the collection $\mathcal{U}=(V_{x}|x\in U)$. Since $\overline{U}$ is compact and $\mathcal{U}$ is an open cover of $\overline{U}$, $\exists$ a finite subcover of $\overline{U}$, say $\{V_{x_1}, \ldots, V_{x_n}\}$. But each $V_{x_i}$ intersects only finitely many sets from $U$.

Therefore, $U\subset \cup_{i=1}^{n}V_{x_i}$ intersects only finitely many sets from $U$.