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).

Theorem.

The centre (http://planetmath.org/GroupCentre) of a Hausdorff topological group is closed.

Proof. Let $Z$ be the centre of a Hausdorff topological group $G$. Let $x\in \overline{Z}$. Then there is a net $(x_{\delta })$ in $Z$ such that $x_{\delta }\to x$. Let $g\in G$. By continuity we have $g{x}_{\delta }{g}^{-1}\to gx{g}^{-1}$. But $g{x}_{\delta }{g}^{-1}=x_{\delta }$, so $g{x}_{\delta }{g}^{-1}\to x$. As $G$ is Hausdorff, these two limits must be the same. So $gx{g}^{-1}=x$, that is, $gx=xg$. Thus $x\in Z$, and we have shown that $\overline{Z}=Z$, 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