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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'') ≥ ...
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]].
# ''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]].