Metric space

Revision as of 22:42, 29 November 2006 by Eyefragment (talk | contribs) (Formatting + brief comments on intuitive metric spaces.)

A metric space is a pair, $(S, d)$ of a set $S$ and a metric $d: S \times S \to \mathbb{R}_{\geq 0}$. The metric $d$ represents a distance function between pairs of points of $S$ which has the following properties:

  • Symmetry ($d(x, y) = d(y, x)$)
  • Non-negativity ($d(x, y) \geq 0$
  • Uniqueness ($d(x, y) = 0$ if and only if $x = y$)
  • Triangle Inequality ($d(x, y) + d(y, z) \geq d(x, z)$ for all points $x, y, z \in S$).

Intuitively, a metric space is a generalization of the distance between two objects (where "objects" can be anything, including points, functions, graphics, or grades). The above properties follow from our notion of distance. Non-negativity stems from the idea that A cannot be closer to B than B is to itself; Uniqueness results from two objects being identical if and only if they are the same object; and the Triangle Inequality corresponds to the idea that a direct path between points A and B should be at least as short as a roundabout path that visits some point C first.

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