every net has a universal subnet

Theorem - (Kelley’s theorem) - Let $X$ be a non-empty set. Every net ${(x_{\alpha })}_{\alpha \in 𝒜}$ 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 $ℱ$ be a section filter for the net ${(x_{\alpha })}_{\alpha \in 𝒜}$.

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

$$(\alpha ,U)\le (\beta ,V)⟺\{\begin{array}{cc}\alpha \le \beta \hfill & \\ V\subseteq U\hfill & \end{array}$$

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

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

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

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

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

As $ℱ$ is a filter, $A\cap S(\alpha )\ne \mathrm{\varnothing }$ and so there exists $\beta$ with $\alpha \le \beta$ such that $x_{\beta }\in A$. Hence, $(\beta ,A)\in 𝒟$.

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

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

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

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

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

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