Editing Directed set (section)

===Directed towards a point===

If <math>x_0</math> is a [[real number]] then the set <math>I := \R \backslash \lbrace x_0 \rbrace</math> can be turned into a directed set by defining <math>a \leq_I b</math> if <math>\left|a - x_0\right| \geq \left|b - x_0\right|</math> (so "greater" elements are closer to <math>x_0</math>). We then say that the reals have been '''directed towards <math>x_0.</math>''' This is an example of a directed set that is {{em|neither}} [[Partial order|partially ordered]] nor [[Total order|totally ordered]]. This is because [[Antisymmetric relation|antisymmetry]] breaks down for every pair <math>a</math> and <math>b</math> equidistant from <math>x_0,</math> where <math>a</math> and <math>b</math> are on opposite sides of <math>x_0.</math> Explicitly, this happens when <math>\{a, b\} = \left\{x_0 - r, x_0 + r\right\}</math> for some real <math>r \neq 0,</math> in which case <math>a \leq_I b</math> and <math>b \leq_I a</math> even though <math>a \neq b.</math> Had this preorder been defined on <math>\R</math> instead of <math>\R \backslash \lbrace x_0 \rbrace</math> then it would still form a directed set but it would now have a (unique) [[greatest element]], specifically <math>x_0</math>; however, it still wouldn't be partially ordered. This example can be generalized to a [[metric space]] <math>(X, d)</math> by defining on <math>X</math> or <math>X \setminus \left\{x_0\right\}</math> the preorder <math>a \leq b</math> if and only if <math>d\left(a, x_0\right) \geq d\left(b, x_0\right).</math>