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===As a metric space=== [[File:Absolute difference.svg|thumb|The [[metric space|metric]] on the real line is [[absolute difference]].]] [[File:Epsilon Umgebung.svg|thumb|An {{math|''Ξ΅''}}-[[Ball (mathematics)|ball]] around a number {{math|''a''}}]] The real line forms a [[metric space]], with the [[distance function]] given by absolute difference: : <math>d(x, y) = |x - y|.</math> The [[metric tensor]] is clearly the 1-dimensional [[Euclidean metric]]. Since the {{mvar|n}}-dimensional Euclidean metric can be represented in matrix form as the {{mvar|n}}-by-{{mvar|n}} identity matrix, the metric on the real line is simply the 1-by-1 identity matrix, i.e. 1. If {{math|''p'' β '''R'''}} and {{math|''Ξ΅'' > 0}}, then the {{mvar|Ξ΅}}-[[Ball (mathematics)|ball]] in {{math|'''R'''}} centered at {{mvar|p}} is simply the open [[Interval (mathematics)|interval]] {{math|(''p'' β ''Ξ΅'', ''p'' + ''Ξ΅'')}}. This real line has several important properties as a metric space: * The real line is a [[complete metric space]], in the sense that any [[Cauchy sequence]] of points converges. * The real line is [[path-connected]] and is one of the simplest examples of a [[geodesic metric space]]. * The [[Hausdorff dimension]] of the real line is equal to one.
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