every net has a universal subnet
Theorem - (Kelley’s theorem) - Let be a non-empty set. Every net in has a universal subnet (http://planetmath.org/Ultranet). That is, there is a subnet such that for every either the subnet is eventually in or eventually in .
Proof : Let be a section filter for the net .
Let . is a directed set under the order relation given by
The map defined by is order preserving and cofinal. Therefore there is a subnet of associated with the map (that is, ).
We now prove that is a net.
Let . We have that is frequently in or frequently in .
Suppose is frequently in .
Let and . We have that by definition of section filter.
As is a filter, and so there exists with such that . Hence, .
As is frequently in , there exists with such that .
Also, is in , and therefore, in . So .
We conclude that for every . Therefore, a filter in . As is a maximal filter we conclude that , and consequently, .
We can now see that for every with , is in and so is in . Therefore, is eventually in .
Remark: If is frequently in , by an analogous we can conclude that it is eventually in .
This proves that is a subnet of .
|Title||every net has a universal subnet|
|Date of creation||2013-03-22 17:25:16|
|Last modified on||2013-03-22 17:25:16|
|Last modified by||asteroid (17536)|