If x_0 is a real number then the set I := \R \backslash \lbrace x_0 \rbrace can be turned into a directed set by defining a \leq_I b if \left|a - x_0\right| \geq \left|b - x_0\right| (so "greater" elements are closer to x_0). We then say that the reals have been directed towards x_0. This is an example of a directed set that is partially ordered nor totally ordered. This is because antisymmetry breaks down for every pair a and b equidistant from x_0, where a and b are on opposite sides of x_0. Explicitly, this happens when \{a, b\} = \left\{x_0 - r, x_0 + r\right\} for some real r \neq 0, in which case a \leq_I b and b \leq_I a even though a \neq b. Had this preorder been defined on \R instead of \R \backslash \lbrace x_0 \rbrace then it would still form a directed set but it would now have a (unique) greatest element, specifically x_0; however, it still wouldn't be partially ordered. This example can be generalized to a metric space (X, d) by defining on X or X \setminus \left\{x_0\right\} the preorder a \leq b if and only if d\left(a, x_0\right) \geq d\left(b, x_0\right).
