# universal nets in compact spaces are convergent

A universal net $(x_{\alpha})_{\alpha\in\mathcal{A}}$ in a compact space $X$ is convergent.

Proof : Suppose by contradiction that $(x_{\alpha})_{\alpha\in\mathcal{A}}$ was not convergent. Then for every $x\in X$ we would find neighborhoods $U_{x}$ such that

 $\forall_{\alpha\in\mathcal{A}}\;\;\;\exists_{\alpha\leq\alpha_{0}}\;\;\;x_{% \alpha_{0}}\notin U_{x}$

The collection of all this neighborhoods cover $X$, and as $X$ is compact, a finite number $U_{x_{1}},U_{x_{2}},\dots,U_{x_{n}}$ also cover $X$.

The net $(x_{\alpha})_{\alpha\in\mathcal{A}}$ is not eventually in $U_{x_{k}}$ so it must be eventually in $X-U_{x_{k}}$ (because it is a net). Therefore we can find $\alpha_{k}\in\mathcal{A}$ such that

 $\forall_{\alpha_{k}\leq\alpha}\;\;\;x_{\alpha}\in X-U_{x_{k}}$

Because we have a finite number $\alpha_{1},\alpha_{2}\dots,\alpha_{n}\in\mathcal{A}$ we can find $\gamma\in\mathcal{A}$ such that $\alpha_{k}\leq\gamma$ for each $1\leq k\leq n$.

Then $x_{\gamma}\in X-U_{x_{k}}$ for all $k$, i.e. $x_{\gamma}\notin U_{x_{k}}$ for all $k$. But $U_{x_{1}},U_{x_{2}},\dots,U_{x_{n}}$ cover $X$ and thus we have a contradiction. $\square$

Title universal nets in compact spaces are convergent UniversalNetsInCompactSpacesAreConvergent 2013-03-22 17:31:29 2013-03-22 17:31:29 asteroid (17536) asteroid (17536) 4 asteroid (17536) Theorem msc 54A20