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 $gx_{\\delta}g^{-1}\\to gxg^{-1}$ .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 $\\overline{Z}=Z$ , as required.

单词	ExampleOfAProofUsingNets
释义	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 $gx_{\\delta}g^{-1}\\to gxg^{-1}$ .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 $\\overline{Z}=Z$ , as required.
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