, one can define its "strict part" and "indifference part" as follows: if and only if and not . if and only if and . Given a strict preference relation , one can define its "weak part" and "indifference part" as follows: if and only if not ; if and only if not and not . For every option , we define the contour sets at A: Given a weak preference relation , the weak upper contour set at A is the set of all options that are at least as good as A: . The weak lower contour set at A is the set of all options that are at most as good as A: . A weak preference relation is called continuous if its contour sets are topologically closed. Similarly, given a strict preference relation , the strict upper contour set at A is the set of all options better than A: , and the strict lower contour set at A is the set of all options worse than A: . A strict preference relation is called continuous if its contour sets are topologically open. Sometimes, the above continuity notions are called semicontinuous, and a is called continuous if it is a closed subset of . A preference-relation is called: Countable - if the set of equivalence classes of the indiffference relation is countable. Separable - if there exists a countable subset such that for every pair , there is an element that separates them, that is, (an analogous definition exists for weak relations). As an example, the strict order ">" on real numbers is separable, but not countable. Utility functions A utility function is a function . A utility function u is said to represent a strict preference relation , if . A utility function u is said to represent a weak preference relation , if . Complete preference relations Debreu proved the existence of a contiuous representation of a weak preference relation satisfying the following conditions: Reflexive and Transitive; Complete, that is, for every two options A, B in X, either or or both; For all , both the upper and the lower weak contour sets are topologically closed; The space X is second-countable. This means that there is a countable set S of open sets, such that every open set in X is the union of sets of the class S. Second-countability is implied by