example of a proof using nets
In this entry we will give a simple example of how nets can be used to prove topological theorems. The proof will make use of some of the basic properties of nets listed in the parent entry (http://planetmath.org/Net).
The centre (http://planetmath.org/GroupCentre) of a Hausdorff topological group is closed.
Proof. Let be the centre of a Hausdorff topological group . Let . Then there is a net in such that . Let . By continuity we have . But , so . As is Hausdorff, these two limits must be the same. So , that is, . Thus , and we have shown that , as required.
|Title||example of a proof using nets|
|Date of creation||2013-03-22 17:11:25|
|Last modified on||2013-03-22 17:11:25|
|Last modified by||yark (2760)|