Nets and Convergence: Why directed indices? Please do read carefully (I know Nets-Topology-Filters and their interrelations!!!)
1.) Why do we require nets to be indexed by directed sets (apart from it simply works compared to filters and topology). Is there a reason w.r.t. to the notion of convergence?
So far there are those hints:


*

*Notion of a good Direction

*Uniqueness of Limit


2.) Moreover, why does it make sense to say that the following net converges even though it grows arbitrarily large on one side:
$$\Lambda:=\{1,2\}\times\mathbb{N}:(i,n)\leq(i',n') :\Leftrightarrow i\leq i',n\leq n'\\ x_{(1,n)} :=n, x_{(2,n)} :=0$$

Thanks in advance. Cheers, Alex

1) For convergence you obviously need some sense of direction. Sequences give a simple of direction (towards $\infty$) but it turns out this is not good enough for dealing with arbitrary topological spaces. So people generalized sequences to nets by using arbitrary directed sets. There was no a priori reason this would work for arbitrary spaces either but it turned out it does work. Filters are equally good and are equivalent to nets.
2) The subnet $y$, given by restricting to the top row, is cofinal in $x$ and it obviously converges to $0$, so $x \to 0$ as well. Informally, the net eventually goes up. More formally, for every $(a,n) \in x$, there is $(b, m) \in y$ such that $(b, m) \geq (a,n)$. Just take $(b, m) := (2, n)$. If you reverted the vertical arrows, the resulting net would go bottom and wouldn't converge.
