I'm trying to prove that R with the usual topology is not compact.
The Attempt at a Solution
According to the solutions, there are two "simple" counterexamples of open coverings that do not contain finite subcoverings: (-n, n) and (n, n+2). Of course finding just one counter-example is sufficient to show that R is not compact. However, it is not clear to me why those are not compact (and something like (0, n) would be compact?).
Couldn't I also use a counter-example where each subcovering of R covers exactly one element of R (i.e. the intersection of the subcovering and R has one element), then observe that R has infinitely many elements and therefore must have infinitely many subcoverings? Having one open covering of A with infinitely many subcoverings would then violate compactness.
edit: I realized why the latter doesn't work - not all elements of R are open sets. I'm still not clear why the subcoverings in the solutions are not compact, though.