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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).

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  • $\begingroup$ Did you realize that the space itself is much much more natural in some sense than the natural numbers - the natural numbers have nothing to do with the space a priori. It is just a bad incident that these occur so natural to us as they provide a simple example of a directed set. But apart from a simple example that's it. $\endgroup$ – C-Star-W-Star Dec 20 '14 at 17:24
  • $\begingroup$ @Freeze_S Indeed, I'm starting to realize that the natural numbers are probably too good to serve as examples. I found a 1978 paper by Saks which characterizes compact spaces as the spaces $S$ for which all nets indexed by cardinals $\mathfrak{m}\leq|S|$ converge relatively to all/some tail ultrafilter (using the generalized notion of convergence along an ultrafilter). I was hoping that the character of the space would have a bigger role when dealing to convergence, but apparently that's not the case. Thank you for your comment. $\endgroup$ – Questioner Dec 22 '14 at 18:04
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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.)

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Let's consider the space $(ω_1 + 1)$, i.e. the transfinite convergent sequence of length $ω_1$. What would your natural directed set be here? You need both $ω$ and $ω_1$ types. So I don't see anything better than $ω × ω_1$ here.

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    $\begingroup$ I understand your point. We'd need $\omega$ to take care of nets converging to points before $\omega_1$ and $\omega_1$ to take care of $\omega_1$ itself. $\endgroup$ – Questioner Dec 20 '14 at 15:28

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