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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.

Martin Brandenburg
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    Did you check in John L. Kelley General Topology? He coined the term. –  Apr 10 '21 at 13:48
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    If you have a net $(x_p)$ with entries in $A_1 \cup A_2$, you can define the sets $P_i = { p \in P \mid x_p \in A_i }$. It is not clear to me why one of these sets should be cofinal directed subset of $P$. – Paul Frost Apr 10 '21 at 13:54
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    You can define a topology from your $\to$, but $\to$ may not coincide with convergence with respect to that topology. – Eric Wofsey Apr 10 '21 at 14:05
  • @EricWofsey Why is that? After all, the topologies have the same closed subsets. – Martin Brandenburg Apr 10 '21 at 14:40
  • @PaulFrost We have a convergent net $(x_p) \to x$ with $x_p \in A \cup B$. When $ x \in A$, we are done. Otherwise, for all $p \in P$ there is some $q > p$ with $x_q \in B$ (since otherwise the subnet indexed by $P_{>p}$ lies in $A$ and converges to $x$, hence $x \in A$). So we get a subnet in $B$. Since $B$ is closed, $x \in B$. – Martin Brandenburg Apr 10 '21 at 14:42
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    For instance, if $\to$ is the empty relation, the topology you get is the discrete topology. But the convergence relation of the discrete topology is not empty, since constant nets converge. – Eric Wofsey Apr 10 '21 at 14:49
  • Thanks! I have added an additional requirement in my post - namely that $\to$ is equal to the net convergence relation from the induced topology (which should be equivalent to: $\to$ is the net convergence relation of some topology). I assumed (wrongly) that this is automatic. – Martin Brandenburg Apr 10 '21 at 15:28
  • @ChrisCuster Thanks! Theorem 9 on p. 74 answers my question. You can post this as an answer, if you want. – Martin Brandenburg Apr 10 '21 at 15:32
  • Don't you have set-theoretic issues? The nets in $X$ form a proper class. – Paul Frost Apr 10 '21 at 16:11
  • @PaulFrost The relation is a class relation (i.e. a formula in two variables). – Martin Brandenburg Apr 10 '21 at 16:44
  • Kelley’s book has a complete characterisation. – Henno Brandsma Apr 10 '21 at 20:47
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    @MartinBrandenburg: It’s probably worth noting that there are two other definitions of subnet, Kelley’s, which is a bit less restrictive, and one due to Aarnes & Andenæs that is even more general than Kelley’s and has the nicest properties of the three. They should be equivalent for your purposes here, however. – Brian M. Scott Apr 10 '21 at 21:04
  • @BrianM.Scott Thanks for pointing this out! – Martin Brandenburg Jun 20 '21 at 11:51
  • @MartinBrandenburg: You’re welcome! – Brian M. Scott Jun 20 '21 at 20:22

2 Answers2

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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.

Martin Brandenburg
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  • By the way, the "Locality" axiom is stated in a double negated form in Kelley's book, which is not necessary at all. – Martin Brandenburg Apr 11 '21 at 11:34
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    What you call "locality" other texts call "topological"; it's a ncessary condition for a net convergence to be induced by a topology. E.g. measure theoretic net convergence does not obey it e.g. There is quite a literature on (topological or not) convergence spaces. The iterated limit condition is the have an idempotent closure operator. – Henno Brandsma Apr 11 '21 at 22:04
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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.