Topology axioms in terms of net convergence

general-topologynetsreference-request

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.

Best Answer

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.

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