I'm just learning about nets.
It occurs to me that sometimes $x_{\alpha} \to x$ doesn't tell us much at all.
Consider the directed set $J:= \{(U,x)\in \mathcal{P}(X) \times X: x \in U \}$, $(U,x) \preceq (V,y) \iff V \subseteq U$, with net $f: J \to X : (U,x) \mapsto x$. (Note that the relation $\preceq$ on $J$ is not antisymmetric.) If there are disjoint nonempty sets in $\mathcal{P}(X)$, we will need to choose $f(\emptyset)$. I believe that $x_{\alpha}$ will always converge to the point $f(\emptyset)$. So really convergence in this case is arbitrary.
I guess this shouldn't really bother me. There's no reason why a net (or a sequence, for that matter) should provide any information about the space. It seems our net is too big for convergence to mean much. On the other extreme, if $J$ is a singleton set, obviously the convergence tells us nothing useful.
Question: Is the net I mention a useful construction? Is there anything I can learn from the above considerations?
The arbitrariness of the convergence comes from the fact that there is what I'll call a final element in $J$. Define $\,\operatorname{Fin}(J):= \{j \in J$ such that $\forall k \neq j \,\,\,\, k \not\succeq j\}$. If $j\in \,\operatorname{Fin}(J)$, any net $f:J \to X$ will always converge $f(j)$ (I claim $ \# (\operatorname{Fin}(J)) \leq 1$ for $J$ a directed set). Another example would be if $J$ were a finite subset of the integers, with the usual order. Then of course any net $f: J \to X$ would converge to the image of the greatest integer in the set.
Convergence of a net can be interesting if $\operatorname{Fin}(J)$ is empty.
Question: is there a canonical name in order theory for what I've called $\text{Fin}(J)$?
Edit: as Miha points out, $\text{Fin}(J)$ is exactly the set of maximal elements.