THE HEINE-BOREL THEOREM

The Heine–Borel TheoremThe interval [0, 1] is compact. Compactness is about open sets. Think of an open set as like a c oud: it has no edge, you can never discern the outermost water drop lets. On the real line, the open interval (0 , 1) is defined to include all real numbers from, but not including, 0 up to,but not including, 1. More generally a subset X of n-dimensional real space, Rn, is open if, for any x in X, there is a distance ǫ > 0 small enough so that x can be surrounded by an-dimensional ball of radiusǫ which lies entirely within X. A set isclosedif its complement is open, and the paradigm of a closed set isR\ (