How do you axiomatize topology via nets? Let $X$ be a set and let ${\mathcal N}$ be a collection of nets on $X.$
I've been told by several different people that ${\mathcal N}$ is the collection of convergent nets on $X$ with respect to some topology if and only if it satisfies some axioms.  I've also been told these axioms are not very pretty.
Once or twice I've tried to figure out what these axioms might be but never came up with anything very satisfying.  Of course one could just recode the usual axioms regarding open sets as statements about nets and then claim to have done the job.  But, come on, that's nothing to be proud of.
Has anyone seen topology axiomatized this way?  Does anyone remember the rules?
 A: (too long for a comment to Pete's answer)
Garrett Birkhoff was my Ph.D. advisor.  Let me provide a few remarks
of a historical nature.
From a 25-year-old Garrett Birkhoff we have: Abstract 355, "A new
definition of limit" Bull. Amer. Math. Soc. 41 (1935) 636.
(Received September 5, 1935)
According to the report of the meeting (Bull. Amer. Math. Soc. 42
(1936) 3) the paper was delivered at the AMS meeting in New York on
October 26, 1935.
In the abstract we find what would nowadays be called convergence
of a filter base.  (See also Definition 4 in Birkhoff's 1937 paper.)
Birkhoff remarked to me once that Bourbaki never acknowledged his
(Birkhoff's) priority.
It seems that some time after Birkhoff's talk, his father (G. D. Birkhoff)
remarked that it reminded him of a paper of Moore and Smith.  So
young Garrett read Moore and Smith, and in the end adopted their
system for the subsequent paper, calling it "Moore-Smith convergence
in general topology".  Since that Annals of Mathematics paper was
received April 27, 1936, one can only imagine young Garrett working
furiously for 6 months converting his previous filter-base material
into the Moore-Smith setting!
A: Yes.  This is given in Kelley's General Topology.  (Kelley was one of the main mathematicians who developed the theory of nets so that it would be useful in topology generally rather than just certain applications in analysis.)
In the section "Convergence Classes" at the end of Chapter 2 of his book, Kelley lists the following axioms for convergent nets in a topological space $X$
a) If $S$ is a net such that $S_n = s$ for each $n$ [i.e., a constant net], then $S$ converges to $s$.
b) If $S$ converges to $s$, so does each subnet.
c) If $S$ does not converge to $s$, then there is a subnet of $S$, no subnet of which converges to $s$.
d) (Theorem on iterated limits): Let $D$ be a directed set.  For each $m \in D$, let $E_m$ be a directed set, let $F$ be the product $D \times \prod_{m \in D} E_m$ and for $(m,f)$ in $F$ let $R(m,f) = (m,f(m))$.  If $S(m,n)$ is an element of $X$ for each $m \in D$ and $n \in E_m$ and $\lim_m \lim_n S(m,n) = s$, then $S \circ R$ converges to $s$.
He has previously shown that in any topological space, convergence of nets satisfies a) through d).  (The first three are easy; part d) is, I believe, an original result of his.)  In this section he proves the converse: given a set $S$ and a set $\mathcal{C}$ of pairs (net,point) satisfying the four axioms above, there exists a unique topology on $S$ such that a net $N$ converges to $s \in X$ iff $(N,s) \in \mathcal{C}$.
I have always found property d) to be unappealing bordering on completely opaque, but that's a purely personal statement.
Addendum: I would be very interested to know if anyone has ever put this characterization to any useful purpose.  A couple of years ago I decided to relearn general topology and write notes this time.  The flower of my efforts was an essay on convergence in topological spaces that seems to cover all the bases (especially, comparing nets and filters) more solidly than in any text I have seen.
http://alpha.math.uga.edu/~pete/convergence.pdf
But "even" in these notes I didn't talk about either the theorem on iterated limits or (consequently) Kelley's theorem above: I honestly just couldn't internalize it without putting a lot more thought into it.  But I've always felt/worried that there must be some insight and content there...
