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 Space:  Examples 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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