I'm just learning how to work with nets. I'm attempting the proof that $X$ compact $\implies$ every net in $X$ has a convergent subnet, and I wonder if I'm overcomplicating it.
Suppose $\langle x_i \rangle _{i \in I}$ is a net in $X$. Define $F_i\subset X$ as $F_i := \operatorname{Cl} \left(\{ x_j: j \succeq i \} \right)$. Observe that $\{ F_i \}$ has the finite intersection property, because given $\cap_{k=1}^n F_{i_k}$, take $i^*:= \operatorname{Join}(i_1, \dotsc, i_n)$ and then $x_{i^*} \in \cap_{k=1}^n F_{i_k}$. Now by compactness, there exists $x\in \cap_{i \in I}F_i$.
It seems clear that the net should "return frequently" to each neighborhood of $x$, and so we can define a convergent subnet. But my construction of the subnet became somewhat involved, and I wondered if there was a simpler way.
I have to find a directed set $J$ and a function $g: J \to I$ such that $j_1 \succeq j_2 \implies g(j_1) \succeq g(j_2)$ and $g(J)$ is cofinal in $I$. What I chose for $J$ were tagged neighborhoods $\mathcal{O}$ of $x$, "tagged" with an element $i \in I$ such that $x_i \in \mathcal{O}_i$. Now we can order $J$ by $\mathcal{O}_{i_1} \succeq \mathcal{O}_{i_2}$ iff $\mathcal{O}_{i_1} \subseteq \mathcal{O}_{i_2}$ and $i_1 \succeq i_2$. Now define the function $g:J \to I: \mathcal{O}_i \mapsto i$. (The tags have allowed the same neighborhood be considered as different elements in $J$ based on their various tags.)
I claim $J$ is a directed set with well-defined join, that $g(J)$ is cofinal in $I$, and that it defines a subnet that converges to $x$.
Does anyone have a less complicated way to construct the convergent subnet?
