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 groupMathworldPlanetmath is closed.

Proof. Let Z be the centre of a Hausdorff topological group G. Let xZ¯. Then there is a net (xδ) in Z such that xδx. Let gG. By continuity we have gxδg-1gxg-1. But gxδg-1=xδ, so gxδg-1x. As G is Hausdorff, these two limits must be the same. So gxg-1=x, that is, gx=xg. Thus xZ, and we have shown that Z¯=Z, as required.

Title example of a proof using nets
Canonical name ExampleOfAProofUsingNets
Date of creation 2013-03-22 17:11:25
Last modified on 2013-03-22 17:11:25
Owner yark (2760)
Last modified by yark (2760)
Numerical id 4
Author yark (2760)
Entry type Example
Classification msc 22A05