I'm studying some things related to ultrafilters on metric and topological spaces and trying to prove theorem in a general setting, so the following question came to my mind.
Let $S$ be a topological space. Does there exist a natural directed set $(M,\leq)$ for which the nets $\left\{x_\lambda\right\}_{\lambda\in M}$ describe the topology of $S$?
By "natural", I mean that the directed set associated to a metric, or first countable, space is $\mathbb{N}$, for example. If we let $\mathcal{U}_x$ denote the collection of open neighbourhoods of a point $x\in S$ ordered by reverse inclusion ($U\leq V\iff V\subseteq U$), then $M=\prod_{x\in S}\mathcal{U}_x$ with the product order is sufficient to describe the topology of $S$ with nets (I believe), but it is not exactly "natural".
Most of the proofs I know of theorems which describe the topology of $S$ using nets usually consider, at some point, a product directed set. When we are dealing with first countable spaces, we usually consider not the product directed set $\mathbb{N}\times\mathbb{N}$, but the diagonal $\left\{(n,n):n\in\mathbb{N}\right\}$, which is in fact cofinal in $\mathbb{N}\times\mathbb{N}$ (see here, for example).
Note that nets can be seen as special filters.
Hausdorff's approach to topology uses neighborhood filters: $\mathcal{N}_x$
Most commonly these are constructed by neighborhood bases: $\mathcal{B}_x$
So one has: $\mathcal{N}_x:=\uparrow\mathcal{B}_x:=\{N:\exists B\in\mathcal{B}_x:B\subseteq N\}$
Now, the topology of metric spaces is easily defined in this way: $B_\varepsilon(x)$
(Precisely, it is defined via a uniform structere constructed via a basis.)