of open sets is said to be an open cover of set if the union of these sets is a superset of . This open cover is said to have a finite subcover if a finite subcollection of the could be found that also covers . Definition. A set in a topological space is compact if every open cover of has a finite subcover. Compact sets are well-behaved with respect to properties like convergence and continuity. For instance, any Cauchy sequence in a compact metric space is convergent. As another example, the image of a compact metric space under a continuous map is also compact. Continuity A function from the set of real numbers to the real numbers can be represented by a graph in the Cartesian plane; such a function is continuous if, roughly speaking, the graph is a single unbroken curve with no "holes" or "jumps". There are several ways to make this intuition mathematically rigorous. Several definitions of varying levels of generality can be given. In cases where two or more definitions are applicable, they are readily shown to be equivalent to one another, so the most convenient definition can be used to determine whether a given function is continuous or not. In the first definition given below, is a function defined on a non-degenerate interval of the set of real numbers as its domain. Some possibilities include , the whole set of real numbers, an open interval or a closed interval Here, and are distinct real numbers, and we exclude the case of being empty or consisting of only one point, in particular. Definition. If is a non-degenerate interval, we say that is continuous at if . We say that is a continuous map if is continuous at every . In contrast to the requirements for to have a limit at a point , which do not constrain the behavior of at itself, the following two conditions, in addition to the existence of , must also hold in order for to be continuous at : (i) must be defined at , i.e., is in the domain of ; and (ii) as . The definition above actually applies to any domain that does not contain an isolated point, or equivalently, where every is a limit point of . A more general definition applying to with a general domain is the following: Definition. If is an arbitrary subset