While studying some basic analysis/topology I have come across the proof regarding that the finite union of compact sets is compact using the definition of compactness.
The proofs I have read all basically follow this:
For each compact set choose a finite subcover. The union of those subcovers will be finite and cover the union of the compact sets.
Alright, not a big deal.
However, I think I may have a misconception regarding my notion of a compact set.
When proving a compact set is indeed compact must I not show that EVERY open cover has a finite subcover?
The aforementioned proof, to me, seams as though it is only considering one possible option for an open cover.
I am not doubting the validity of the proof, instead I am looking for some clarification as to why we do not consider the possibility of other open covers.