sentences:  Indeed, the Heine–Borel definition of compactness—that every covering of a space by open sets admits a finite subcovering—is relatively recent.More popular in the 19th and early 20th centuries was the Bolzano–Weierstrass criterion that every sequence admits a convergent subsequence, now called sequential compactness.These conditions are equivalent for metrizable spaces, but neither one implies the other in the class of all topological spaces.