Two metrics and defined on a space are called equivalent if they induce the same metric topology on . This is the case iff, for every point of , every ball with center at defined with respect to :
(1)
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contains a ball with center with respect to :
(2)
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and conversely.
Every metric on has uncountably many equivalent metrics. For every positive real number , a "scaled" metric can be defined such that for all ,
(3)
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In fact, for all :
(4)
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Another metric equivalent to is defined by
(5)
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for all . In fact,
(6)
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and
(7)
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In the Euclidean plane , the metric
(8)
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with circular balls can be defined in addition to the Euclidean metric. An equivalent more general metric for all positive real numbers and can be defined as
(9)
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with elliptic balls, and the taxicab metric
(10)
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can be defined with square "balls." All these are equivalent to the Euclidean metric.