Kleiman's criterion implies that ampleness is an open condition on the class of an R-divisor (an R-linear combination of Cartier divisors) in , with its topology based on the topology of the real numbers. (An R-divisor is defined to be ample if it can be written as a positive linear combination of ample Cartier divisors.) An elementary special case is: for an ample divisor H and any divisor E, there is a positive real number b such that is ample for all real numbers a of absolute value less than b. In terms of divisors with integer coefficients (or line bundles), this means that nH + E is ample for all sufficiently large positive integers n. Ampleness is also an open condition in a quite different sense, when the variety or line bundle is varied in an algebraic family. Namely, let be a proper morphism of schemes, and let L be a line bundle on X. Then the set of points y in Y such that L is ample on the fiber is open (in the Zariski topology). More strongly, if L is ample on one fiber , then there is an affine open neighborhood U of y such that L is ample on over U. Kleiman's other characterizations of ampleness Kleiman also proved the following characterizations of ampleness, which can be viewed as intermediate steps between the definition of ampleness and numerical criteria. Namely, for a line bundle L on a proper scheme X over a field, the following are equivalent: L is ample. For every (irreducible) subvariety of positive dimension, there is a positive integer r and a section which is not identically zero but vanishes at some point of Y. For every (irreducible) subvariety of positive dimension, the holomorphic Euler characteristics of powers of L on Y go to infinity: as . Generalizations Ample vector bundles Robin Hartshorne defined a vector bundle F on a projective scheme X over a field to be ample if the line bundle on the space of hyperplanes in F is ample. Several properties of ample line bundles extend to ample vector bundles. For example, a vector bundle F is ample if and only if high symmetric powers of F kill the cohomology of coherent sheaves for all . Also, the Chern class of an ample vector bundle has positive degree on every r-dimensional subvariety of X, for . Big line bundles