Alexander's subbase theorem

Alexander's subbase theorem provides a way to verify the compactness of a topological space by examining its subcollections of open sets. Specifically, if a space's open sets are generated by a subcollection such that every open cover from this subcollection has a finite subcover, then the entire space is compact. In simpler terms, instead of checking every open cover, you can focus on a smaller, strategic set of open sets (the subbase) to determine whether the entire space is compact, making the verification process more manageable.