I've been struggling to find a solution to this problem:
For which compact sets can you set an upper bound on the number of sets in a subcover of an open cover.
My understanding is that I need to show that there are compact sets that cannot have infinite subcovers. This is different from showing that every open cover of a compact set has a finite subcover. Please let me know if I am interpreting this problem correctly and I'd appreciate any help coming up with an answer. Thanks!
Best Answer
By definition (of compactness), given any compact space $X$, any open cover of $X$ has a finite subcover. In general, however, one cannot say before one has an open cover how many open sets one might need to make a subcover.
For example, for any $\epsilon > 0$, the set of $\epsilon$-balls in $[0, 1]$ is an open cover, and so admits some finite subcover. One needs $\approx \frac{1}{2 \epsilon}$ of these balls to cover the entire interval, and this number can be made arbitrarily large my making $\epsilon$ sufficiently small. So, even though we can always construct finite open subcovers of $[0, 1]$, we cannot say a priori how many subsets we'll need.
The question is asking for a condition on the compact set $X$ for which this issue doesn't occur, that is, for which you can say before you get an open cover of $X$ (at most) how many open sets you'll need to make a subcover.