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. 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

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 universal 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, $\mathcal{F} \cup \{E\}$ generates 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 argument we can conclude that it is eventually in $X-E$ .

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