Upper bound on number of sets in a sub cover? An exercise in Strichartz asks "For which compact sets can you set an upper bound on the number of sets in a sub cover on an open cover?"
This doesn't make sense to me. How could there ever be an upper bound? If we are allowed to take any open cover that we want, then why not choose the open cover $\{ (-n,n): n\in \mathbb{N}\}$. This covers the whole line so it covers any compact subset. But $\{(-2n,2n)\}$ is a proper sub cover that has infinitely many elements. So what does Strichartz mean?
It seems to me that this question would make more sense if it were asking for an upper bound on the sizes of minimal sub covers. 

So, given the compact set $X$, and an open cover $\mathcal A$ of it, here is a finite subcover.  What is the minimal size of such a finite subcover of $X$ chosen from $\mathcal A$?  Of course, that minimal size depends on which cover $\mathcal A$ you start with.  Strichartz is asking: could you have a compact set $X$ so that there is a number $N$ such that, for any open cover, there is a subcover of size at most $N$?
