Topology axioms in terms of net convergence

Question

I am looking for the list of axioms of "net convergence" in the language of nets which correspond to the axioms of a topology. (Notice that neither Wikipedia nor nlab seem to answer this question.) Specifically:

Let $X$ be a set. A net in $X$ is defined as a function $P \to X$ from a directed partial order $P$ to $X$. Let $\to$ be a relation from nets in $X$ to elements of $X$, thought of as net convergence. Now let us call $A \subseteq X$ closed if it is closed under net convergence: $$(x_p)_{p\in P} \to x ~ \wedge~ \forall p \in P (x_p \in A) \implies x \in A.$$ Question. What are axioms for $\to$ which guarantee that this is a topology on $X$ such that the notion of net convergence from the topology is exaclty $\to$?

If I am not mistaken, we just need that $\to$ is compatible with subnets: A subnet of $P \to X$ is a composition $Q \to P \to X$ for some cofinal map of partial orders $Q \to P$. We need to require that if a net converges to some element, then every subnet convergences to that element as well.

Then all the axioms of a topology are satisfied: $\emptyset$ is closed since there is no net with values in $\emptyset$ (remember that directed sets are non-empty by definition). The intersection of closed subsets is closed for trivial reasons. Now if $ A,B$ are closed and a net $(x_p)_{p \in P}$ with entries in $A \cup B$ converges to some element $x \in X \setminus A$, then it has a subnet with entries in $B$, thus $x \in B$.

This means that we just need one axiom, which is a bit weird. What I am missing? In particular, I don't see directly how to deduce that a constant net $(x)_{p \in P}$ converges to $x$.

I would appreciate references to the literature. It seems to be a very basic question. But when I look for these kind of characterizations, the texts seem to focus on filters instead.

Answer. The question is answered by Theorem 9 on p. 74 in Kelley's book General topology. Thanks Chris Custer for pointing this out.

Answer 1

The question is answered by Theorem 9 on p. 74 in Kelley's book General topology. Thanks Chris Custer for pointing this out.

The following four axioms are necessary and sufficient (their names are my choices):

• Constant nets. Every constant net converges to its value.
• Subnets. If a net converges to some element, then every subnet converges to that element as well.
• Locality. A net converges to some element when every subnet has a subnet which converges to that element.
• Iterated limits. Let $P$ be a directed set, and let $Q_p$ be a directed set for each $p \in P$. Let $(x_{p,q})$ be a family of elements in $ X$ indexed by $p \in P$ and $q \in Q_p$. Assume that for each $p \in P$ the net $(x_{p,q})_{q \in Q_p}$ converges to some element $x_p \in X$, and that $(x_p)_{p \in P}$ converges to some element $s \in X$. Then the net $(x_{p,f(p)})$ indexed by the product $(p,f) \in P \times \prod_{p \in P} Q_p$ converges to $s$ as well.

Edit. These axioms can be used to define a limit sketch which models $\mathbf{Top}$, see Large limit sketches and topological space objects, Sections 7 and 8.

Answer 2

I had a hunch it might be in Kelley, since I read on Wikipedia or something that he coined the term. Nets generalize the notion of sequence. Filters are an alternative to nets. I remember Henri Cartan with regard to the latter. Each has advantages and disadvantages.

Henno Brandsma told me before that Kelley really sort of pushed the idea of nets, but that was in the $50$'s and in more recent treatments they aren't used as much.

We don't need a first countable space for nets. Also, they have unique limits precisely when the space is Hausdorff.