the following properties (from weaker to stronger): The space X is separable and connected. The relation is separable. The relation is countable. Jaffray gives an elementary proof to the existence of a continuous utility function. Incomplete preference relations Preferences are called incomplete when some options are incomparable, that is, neither nor holds. This case is denoted by . Since real numbers are always comparable, it is impossible to have a representing function u with . There are several ways to cope with this issue. One-directional representation Peleg defined a utility function representation of a strict partial order as a function such that, that is, only one direction of implication should hold. Peleg proved the existence of a one-dimensional continuous utility representation of a strict preference relation satisfying the following conditions: Irreflexive and transitive (which implies that it is asymmetric, that is, is a strict partial order); Separable; For all , the lower strict contour set at A is topologically open; Spacious: if , then the lower strict contour set at A contains the closure of the lower strict contour set at B. This condition is required for incomplete preference relations. For complete preference relations, every relation in which all lower and upper strict contour sets are open, is also spacious. If we are given a weak preference relation , we can apply Peleg's theorem by defining a strict preference relation: if and only if and not . The second condition ( is separable) is implied by the following three conditions: The space X is separable; For all , both lower and upper strict contour sets at A are topologically open; If the lower countour set of A is nonempty, then A is in its closure. A similar approach was taken by Richter. Therefore, this one-directional representation is also called a Richter-Peleg utility representation. Jaffray defines a utility function representation of a strict partial order as a function such that both , and , where the relation is defined by: for all C, and (that is: the lower and upper contour sets of A and B are identical). He proved that, for every partially-ordered space that is perfectly-separable, there exists a utility function that is upper-semicontinuous in any topology stronger than the upper order topology. An analogous statement states the existence of a utility function that is lower-semicontinuous in any topology stronger than the lower order topology. Sondermann defines a utility