If ≿ is a weak order on X,
x ≿ y ⇒ ≿(x) ⊂ ≿(y)†
x ≻ y ⇒ ≻(x) ⊂ ≻(y)†
The principal task of optimization theory and practice is identify the best element(s) in a choice set X, which is usually weakly ordered by some criterion. To identify the best element, optimization theory draws on other properties (linear and metric) of the choice set. Techniques of optimization are explored in chapter 5. To prepare the ground, we next investigate the metric and linear properties of sets in sections 1.3 and 1.4. Before leaving order relations, we touch on the problem of aggregating different orders on a common set.