example of a proof using nets

In this entry we will give a simple exampleof how nets can be used to prove topological theorems.The proof will make use of some of the basic properties of netslisted 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.