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| In the study of [[metric spaces]] in [[mathematics]], there are various notions of two [[metric (mathematics)|metrics]] on the same underlying space being "the same", or '''equivalent'''.
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| In the following, <math>X</math> will denote a non-[[empty set]] and <math>d_{1}</math> and <math>d_{2}</math> will denote two metrics on <math>X</math>.
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| ==Topological equivalence==
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| The two metrics <math>d_{1}</math> and <math>d_{2}</math> are said to be '''topologically equivalent''' if they generate the same [[topology]] on <math>X</math>. The adjective "topological" is often dropped.<ref>Bishop and Goldberg, p. 10.</ref> There are multiple ways of expressing this condition:
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| * a subset <math>A \subseteq X</math> is <math>d_{1}</math>-[[open set|open]] [[if and only if]] it is <math>d_{2}</math>-open;
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| * the [[open ball]]s "nest": for any point <math>x \in X</math> and any radius <math>r > 0</math>, there exist radii <math>r', r'' > 0</math> such that
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| :<math>B_{r'} (x; d_{1}) \subseteq B_{r} (x; d_{2})</math> and <math>B_{r''} (x; d_{2}) \subseteq B_{r} (x; d_{1}).</math>
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| * the [[identity function]] <math>I : X \to X</math> is both <math>(d_{1}, d_{2})</math>-[[continuous function|continuous]] and <math>(d_{2}, d_{1})</math>-continuous.
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| The following are sufficient but not necessary conditions for topological equivalence:
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| * there exists a strictly increasing, continuous, and [[subadditive]] <math>f:\mathbb{R}_{+} \to \mathbb{R}</math> such that <math>d_{2} = f \circ d_{1}</math>.<ref>Ok, p. 127, footnote 12.</ref>
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| * for each <math>x \in X</math>, there exist positive constants <math>\alpha</math> and <math>\beta</math> such that, for every point <math>y \in X</math>,
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| :<math>\alpha d_{1} (x, y) \leq d_{2} (x, y) \leq \beta d_{1} (x, y).</math> | |
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| ==Strong equivalence==
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| Two metrics <math>d_{1}</math> and <math>d_{2}</math> are '''strongly equivalent''' if and only if there exist positive constants <math>\alpha</math> and <math>\beta</math> such that, for every <math>x,y\in X</math>,
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| :<math>\alpha d_{1}(x,y) \leq d_{2}(x,y) \leq \beta d_{1} (x, y).</math> | |
| In contrast to the sufficient condition for topological equivalence listed above, strong equivalence requires that there is a single set of constants that holds for every pair of points in <math>X</math>, rather than potentially different constants associated with each point of <math>X</math>.
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| Strong equivalence of two metrics implies topological equivalence, but not vice versa. An intuitive reason why topological equivalence does not imply strong equivalence is that [[Bounded set#Metric space|bounded sets]] under one metric are also bounded under a strongly equivalent metric, but not necessarily under a topologically equivalent metric.
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| All metrics induced by the [[p-norm]], including the [[euclidean metric]], the [[taxicab metric]], and the [[Chebyshev distance]], are strongly equivalent.<ref>Ok, p. 138.</ref>
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| Even if two metrics are strongly equivalent, not all properties of the respective metric spaces are preserved. For instance, a function from the space to itself might be a [[contraction mapping]] under one metric, but not necessarily under a strongly equivalent one.<ref>Ok, p. 175.</ref>
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| ==Properties preserved by equivalence==
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| * The [[continuous function|continuity]] of a function is preserved if either the domain or range is remetrized by an equivalent metric, but [[uniform continuity]] is preserved only by strongly equivalent metrics.<ref>Ok, p. 209.</ref>
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| * The [[differentiability]] of a function is preserved if either the domain or range is remetrized by a strongly equivalent metric.<ref>Cartan, p. 27.</ref>
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| ==Notes==
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| {{reflist}}
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| ==References==
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| {{refbegin}}
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| * {{cite book
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| | authors = Richard L. Bishop, Samuel I. Goldberg
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| | title = Tensor analysis on manifolds
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| | year = 1980
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| | publisher = Dover Publications
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| | url = http://books.google.com/books?id=LAuN5-og4jwC
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| }}
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| * {{cite book
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| | author = Efe Ok
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| | title = Real analysis with economics applications
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| | year = 2007
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| | publisher = Princeton University Press
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| | isbn = 0-691-11768-3
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| }}
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| * {{cite book
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| | author = Henri Cartan
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| | title = Differential Calculus
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| | year = 1971
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| | publisher = Kershaw Publishing Company LTD
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| | isbn = 0-395-12033-0
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| }}
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| {{refend}}
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| [[Category:Metric geometry]]
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