Difference between revisions of "Metric space"

Revision as of 19:11, March 20, 2007

A metric space is a pair (X, d) where X is a topological space and d is a metric.

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-A '''metric space''' is a pair (''X'', ''d'') where ''X'' is a [[topological space]] and ''d'' is a function mapping elements in ''M'' to real numbers, such that
+A '''metric space''' is a pair (''X'', ''d'') where ''X'' is a [[topological space]] and ''d'' is a [[Metric (mathematics)|metric]].
-# ''d''(''x'', ''y'') ≥ 0 &nbsp;&nbsp;&nbsp; (''non-negativity'')
-# ''d''(''x'', ''y'') = 0 &nbsp; if and only if &nbsp; ''x'' = ''y'' &nbsp;&nbsp;&nbsp; (''identity'')
-# ''d''(''x'', ''y'') = ''d''(''y'', ''x'')  &nbsp;&nbsp;&nbsp; (''symmetry'')
-# ''d''(''x'', ''z'') ≤ ''d''(''x'', ''y'') + ''d''(''y'', ''z'') &nbsp;&nbsp;&nbsp; (''[[triangle inequality]]'').
-The function ''d'' generalizes the notion of length from ordinary [[Euclidean space]].
 [[Category: Mathematics]]