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?
Best Answer
Yes, this is the standard way. With nets we often see this "product trick" to construct subnets. Also, using reversely ordered neighbourhoods is a common theme as well. You could compare your write-up with mine here, e.g., and see that is essentially the same.