# 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 |