Compactness is one of the hardest things in the undergrad curriculum to teach, I find. It’s very hard for students to grasp that the issue is not “is there a finite collection of open sets that covers K” but rather “for every collection of open sets that covers K, some finite subcollection covers K.” This year I came up with a new metaphor. I asked the students to think about how, when a professor assigns a group project, there’s always a very small subgroup of the students who does all the work. That’s sort of how compactness works! Yes, the professor assigned infinitely many open sets to the project, but actually most of them are not really contributing. And it doesn’t matter if some small set of students could do the project; what matters is that some small group among the students assigned to the project is going to end up doing it! And this seems to happen — my students nodded in agreement — no matter how the professor forms the group.