Metric Space

A metric space is an ordered pair where is a set and is a metric on, i.e., a function

such that for any, the following holds:

  1. (non-negative),
  2. iff (identity of indiscernibles),
  3. (symmetry) and
  4. (triangle inequality) .

The first condition follows from the other three, since:

The function is also called distance function or simply distance. Often, is omitted and one just writes for a metric space if it is clear from the context what metric is used.

Read more about Metric SpaceExamples of Metric Spaces, Open and Closed Sets, Topology and Convergence, Types of Maps Between Metric Spaces, Notions of Metric Space Equivalence, Topological Properties, Distance Between Points and Sets; Hausdorff Distance and Gromov Metric, Product Metric Spaces, Quotient Metric Spaces, Generalizations of Metric Spaces

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