Compactness in Topology

In topology, compactness is a property that generalizes the idea of a set being "bounded" and "closed" from basic geometry. A space is compact if every collection of open sets that covers the space (meaning they together contain the entire space) has a finite subcollection that also covers it. This implies that the space can be "contained" or "covered" efficiently, avoiding the need for infinitely many pieces. Compactness is important because it ensures the space has desirable properties, like continuous functions reaching maximum or minimum values, and it often indicates a form of "completeness" or "finiteness" within an infinite context.