Metric space

From AoPSWiki

A metric space is a pair, of a set and a metric . The metric represents a distance function between pairs of points of which has the following properties:

• Symmetry: for all ,
• Non-negativity: for all ,
• Uniqueness: for all , if and only if
• The Triangle Inequality: for all points ,

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.

Popular metrics

• The Euclidean metric on , with the "usual" meaning of distance

• The Discrete metric on any set

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