TITLE: How do you axiomatize topology via nets?
QUESTION [28 upvotes]: 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?

REPLY [6 votes]: (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!