Heine-Borel theorem

The Heine-Borel theorem states that in finite-dimensional space like the usual 3D world, a set is "compact"—meaning it is both bounded (fits within some finite region) and closed (contains all its boundary points)—if and only if every open cover (a collection of open sets that together cover the set) has a finite subcover (a smaller finite collection that still covers the set). In simple terms, this ensures that such sets are manageable, contain all limit points, and can be neatly covered with finitely many simple building blocks, which is fundamental for various areas in mathematics.