# every net has a universal subnet

Theorem - (Kelley’s theorem) - Let $X$ be a non-empty set. Every net $(x_{\alpha})_{\alpha\in\mathcal{A}}$ in $X$ has a universal  subnet (http://planetmath.org/Ultranet). That is, there is a subnet such that for every $E\subseteq X$ either the subnet is eventually in $E$ or eventually in $X-E$.

Proof : Let $\mathcal{F}$ be a section filter for the net $(x_{\alpha})_{\alpha\in\mathcal{A}}$.

Let $\mathcal{D}=\{(\alpha,U):\alpha\in\mathcal{A}\;,\;U\in\mathcal{F},\;x_{\alpha}% \in U\}$. $\mathcal{D}$ is a directed set  under the order relation given by

 $(\alpha,U)\leq(\beta,V)\Longleftrightarrow\begin{cases}\alpha\leq\beta\\ V\subseteq U\end{cases}$

The map $f:\mathcal{D}\longrightarrow\mathcal{A}$ defined by $f(\alpha,U):=\alpha$ is order preserving and cofinal. Therefore there is a subnet $(y_{(\alpha,U)})_{(\alpha,U)\in\mathcal{D}}$ of $(x_{\alpha})_{\alpha\in\mathcal{A}}$ associated with the map $f$ (that is, $y_{(\alpha,U)}=x_{\alpha}$).

We now prove that $(y_{(\alpha,U)})_{(\alpha,U)\in\mathcal{D}}$ is a net.

Let $E\subseteq X$. We have that $(y_{(\alpha,U)})_{(\alpha,U)\in\mathcal{D}}$ is frequently in $E$ or frequently in $X-E$.

Suppose $(y_{(\alpha,U)})_{(\alpha,U)\in\mathcal{D}}$ is frequently in $E$.

Let $A\in\mathcal{F}$ and $S(\alpha):=\{x_{\beta}:\alpha\leq\beta\}$. We have that $S(\alpha)\in\mathcal{F}$ by definition of section filter.

As $\mathcal{F}$ is a filter, $A\cap S(\alpha)\neq\emptyset$ and so there exists $\beta$ with $\alpha\leq\beta$ such that $x_{\beta}\in A$. Hence, $(\beta,A)\in\mathcal{D}$.

As $(y_{(\alpha,U)})_{(\alpha,U)\in\mathcal{D}}$ is frequently in $E$, there exists $(\gamma,B)\in\mathcal{D}$ with $(\beta,A)\leq(\gamma,B)$ such that $y_{(\gamma,B)}\in E$.

Also, $y_{(\gamma,B)}$ is in $B$, and therefore, in $A$. So $A\cap E\neq\emptyset$.

We conclude that $E\cap A\neq\emptyset$ for every $A\in\mathcal{F}$. Therefore, $\mathcal{F}\cup\{E\}$ a filter in $X$. As $\mathcal{F}$ is a maximal filter we conclude that $E\in\mathcal{F}$, and consequently, $(\gamma,E)\in\mathcal{D}$.

We can now see that for every $(\delta,C)$ with $(\gamma,E)\leq(\delta,C)$, $y_{(\delta,C)}$ is in $C$ and so is in $E$. Therefore, $(y_{(\alpha,U)})_{(\alpha,U)\in\mathcal{D}}$ is eventually in $E$.

Remark: If $(y_{(\alpha,U)})_{(\alpha,U)\in\mathcal{D}}$ is frequently in $X-E$, by an analogous we can conclude that it is eventually in $X-E$.

This proves that $(y_{(\alpha,U)})_{(\alpha,U)\in\mathcal{D}}$ is a subnet of $(x_{\alpha})_{\alpha\in\mathcal{A}}$. $\square$

Title every net has a universal subnet EveryNetHasAUniversalSubnet 2013-03-22 17:25:16 2013-03-22 17:25:16 asteroid (17536) asteroid (17536) 8 asteroid (17536) Theorem msc 54A20 Kelley’s theorem Ultranet