Difference between revisions of "Metric space"
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m (New page: 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 # ''d''(''x'', ''y'') ≥ ...) |
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The function ''d'' generalizes the notion of length from ordinary [[Euclidean space]]. | The function ''d'' generalizes the notion of length from ordinary [[Euclidean space]]. | ||
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Revision as of 19:05, March 20, 2007
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
- d(x, y) ≥ 0 (non-negativity)
- d(x, y) = 0 if and only if x = y (identity)
- d(x, y) = d(y, x) (symmetry)
- d(x, z) ≤ d(x, y) + d(y, z) (triangle inequality).
The function d generalizes the notion of length from ordinary Euclidean space.