example of a proof using nets
ExampleOfAProofUsingNets
2007-06-03 16:07:55
2007-06-03 16:07:55
Example
net
centre
center
Hausdorff
topological group
closed
\usepackage{amsthm}
\newtheorem*{thm*}{Theorem}
\def\closure{\overline}
\PMlinkescapeword{basic}
\PMlinkescapeword{centre}
\PMlinkescapeword{center}
\PMlinkescapeword{limits}
\PMlinkescapeword{properties}
\PMlinkescapeword{simple}
\PMlinkescapeword{theorem}
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 \PMlinkname{parent entry}{Net}.
\begin{thm*}
The \PMlinkname{centre}{GroupCentre} of a Hausdorff topological group is closed.
\end{thm*}
{\bf Proof.}
Let $Z$ be the centre of a Hausdorff topological group $G$.
Let $x\in\closure{Z}$.
Then there is a net $(x_\delta)$ in $Z$ such that $x_\delta\to x$.
Let $g\in G$.
By continuity we have $gx_\delta g^{-1}\to gxg^{-1}$.
But $gx_\delta g^{-1}=x_\delta$, so $gx_\delta g^{-1}\to x$.
As $G$ is Hausdorff, these two limits must be the same.
So $gxg^{-1}=x$, that is, $gx=xg$.
Thus $x\in Z$,
and we have shown that $\closure{Z}=Z$, as required.