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| | {{Distinguish2|[[Winding number]] or [[degree of a continuous mapping]], sometimes called "covering number" or "engulfing number"}} |
| | 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>. |
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| | 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> . |
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| | [[Category:Topology]] |
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Revision as of 22:20, 26 February 2013
Template:Distinguish2
In mathematics, the -covering number of a set , denoted by , in a metric space for some is the minimum number of balls of radius that are needed to cover . For example, when is the -dimensional Euclidean space and is the unit Euclidean ball, .
A related concept is the -packing number which is defined as the maximum number of disjoint balls of radius that fit into .
The covering number and the packing number are related by the following inequalities:
- .
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