Definition Suppose is a metric space with a metric , and suppose is a subset of . Let be a positive real number. A subset is an -net for if, for all , there is an , such that .
For any and , the set is trivially an -net for itself.
Theorem Suppose is a metric space with a metric , and suppose is a subset of . Let be a positive real number. Then is an -net for , if and only if
is a cover for . (Here is the open ball with center and radius .)
Example In with the usual Cartesian metric, the set
is an -net for assuming that .
The above definition and example can be found in , page 64-65.
- 1 G. Bachman, L. Narici, Functional analysis, Academic Press, 1966.
|Date of creation||2013-03-22 13:37:54|
|Last modified on||2013-03-22 13:37:54|
|Last modified by||Koro (127)|