Prove the following.
X is compact if and only if every net in X has a convergent subnet.
I am stuck in proving the (<=) side.
https://dbfin.com/topology/munkres/chapter-3/supplementary-exercises-nets/problem-10-solution/ I did find a solution here, but I think it has an error. The set $K$ of all finite subsets of $J$ should have the empty set as an element in order to have a well-defined partial relation(because two finite subsets of $J$ can be disjoint, but there still should be an element of $K$ that is a subset of both). And in the last part, from $ \beta \subset \{k\}$ it is possible that $\beta $ is the empty set, so it is impossible to conclude that $U \cap B_k \neq 0$.
Best Answer
I think they made a mistake in defining the order relation. You should order finite subset by inclusion instead of reverse inclusion. If you do that then the proof works fine.