Papyrus 41: Difference between revisions

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{{Distinguish2|[[Winding number]] or [[degree of a continuous mapping]], sometimes called "covering number" or "engulfing number"}}
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In mathematics, the '''<math>\varepsilon</math>-covering number''' of a set <math>K</math>, denoted by <math>C(K,\varepsilon)</math>, in a [[metric space]] <math>(X,d)</math> for some <math>\varepsilon>0</math> is the minimum number of balls of radius <math>\varepsilon</math> that are needed to cover <math>K</math>. For example, when <math>(X,d)</math> is the <math>n</math>-dimensional Euclidean space and <math>K</math> is the unit Euclidean ball, <math>C(K,\varepsilon) = O((2/\varepsilon)^n)</math>.


A related concept is the '''<math>\varepsilon</math>-packing number''' which is defined as the maximum number of disjoint balls of radius <math>\varepsilon/2</math> that fit into <math>K</math>.
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The covering number <math>C(K,\varepsilon)</math> and the packing number <math>P(X,\varepsilon)</math> are related by the following inequalities:
: <math>C(K,2\varepsilon) \le P(K,2\varepsilon) \le C(K,\varepsilon)</math> .
 
 
[[Category:Topology]]
 
 
{{topology-stub}}

Latest revision as of 19:41, 18 August 2014

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