Lemma 5.3.9.label Let $X$ be a set, $\angles{x_\alpha}_{\alpha \in A}\subset X$ be a net, and $\fF \subset 2^{X}$ be a filter, then the following are equivalent:
- (1)
For each $F \in \fF$, $\angles{x_\alpha}_{\alpha \in A}$ is in $F$ frequently.
- (2)
There exists an subnet $\angles{x_\beta}_{\beta \in B}\subset \angles{x_\alpha}_{\alpha \in A}$ such that for each $F \in \fF$, $\angles{x_\beta}_{\beta \in B}$ is in $F$ eventually.
Proof. (1) $\Rightarrow$ (2): Let
with the ordering $(\alpha, E) \ge (\beta, F)$ if $\alpha \le \beta$ and $E \supset F$, then $B$ is upward-directed. As $\angles{x_\alpha}_{\alpha \in A}$ is in $F$ frequently for all $F \in \fF$, the mapping $(\alpha, E) \mapsto \alpha$ is upward-directed.
For each $(x, E) \in B$, let $x_{(\alpha, E)}= x_{\alpha}$, then $\anglesn{x_{(\alpha, E)}}_{(\alpha,E) \in B}$ is a subnet of $\angles{x_\alpha}_{\alpha \in A}$.$\square$
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