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A convergence class is defined from nlab

A convergence space is a set $S$ together with a relation $→$ from $ℱS$ to $S$, where $ℱS$ is the set of filters on $S$; if $F→x$, we say that $F$ converges to $x$ or that $x$ is a limit of $F$. This must satisfy some axioms:

  • Centred: The principal ultrafilter $F_x=\{A∣x∈A\}$ at $x$ converges to $x$;
  • Isotone: If $F⊆G$ and $F→x$, then $G→x$;
  • Directed: If $F→x$ and $G→x$, then some filter contained in the intersection $F∩G$ converges to $x$. In light of (2), it follows that $F∩G→x$ itself. (Strictly speaking, the relation should not be called directed unless also every point is a limit of some filter, but this follows from 1.)

A convergence space is topological if it comes from a topology on S.

My questions:

  1. The definition can also be phrased in terms of nets; a net $ν$ converges to $x$ if and only if its eventuality filter (defined as follows) converges to $x$.

    Let $X$ be a set, let $D$ be a directed set, and let $n$ be a function from $D$ to $X$, so that $n$ is a net in $X$. Given a subset $A$ of $X$, $n$ is eventually in $A$ if, for some $i$ in $D$, for each $j≥i$ in $D$, $n_j∈A$. The collection $F_n$ of all those subsets $A$ such that $x$ is eventually in $A$ is a proper filter on $X$, called the eventuality filter of $n$.

    I was wondering how the definition of a convergence space can be equivalently rephrased in terms of nets?

  2. How is the definition of a convergence space related the definition of a convergence class which is defined in terms of nets as following (quoted from Pete Clark at MO):

    In the section "Convergence Classes" at the end of Chapter 2 of General Topology, Kelley lists the following axioms for convergent nets in a topological space $X$

    a) If $S$ is a net such that $Sn=s$ 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 $m∈D$ and $n\in E_m$ and $\lim_m \lim_n S(m,n)=s$, then $S∘R$ 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 $X$ and a set $C$ of pairs (net,point) satisfying the four axioms above, there exists a unique topology on $X$ such that a net $S$ converges to $s∈X$ iff $(S,s)∈C$.

    In particular, Kelley's convergence class is topologizable while nlab's convergence space may not be. If by omitting some of its axioms, will the generalization of convergence class become equivalent to nlab's convergence space?

Thanks and regards!

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1 Answer 1

For a net and its eventuality filter we have that they have exactly the same limits in $X$. Because a net $n$ converges to $p$ iff $n$ is eventually in every neighbourhood of $p$ iff all neighbourhoods of $p$ are in the eventuality filter of $n$ iff the eventuality filter of $n$ converges to $p$.

The eventuality filter is the filter generated by the filterbase of tails of the net: if $n: D \rightarrow X$ is a net (defined on some directed set $D$) then a tail of $n$ is a set of the form $T(j) = \left\{n(i) : i \ge j \right\}$, for some $j \in D$. The tails form a filterbase because the set $D$ is directed. So we have a natural function $F$ that sends a net $n$ to a filterbase $F(n)$, such that $n \rightarrow p$ iff $F(n) \rightarrow p$.

There is also a function going back: if we start with a filterbase $\mathcal{B}$, we can define a directed order on $X \times \mathcal{B}$ by $(x,B) \le (x',B')$ iff $B' \subset B$, (directed of course because $\mathcal{B}$ is a filterbase!) and a natural net $G(\mathcal{B})$ that sends $(x,B)$ to $x$. Then again (check the definitions) we know that $\mathcal{B}$ converges to $p$ (defined as: every neighbourhood of $p$ contains a member of $\mathcal{B}$, or equivalently, the generated filter $\mathcal{F}$ by $\mathcal{B}$, i.e. all supersets of members of the filterbase, converges to $p$) iff the net $G(\mathcal{B})$ converges to $p$.

Also one can check (a bit tedious) that $F(G(\mathcal{B}))= \mathcal{B}$ and reversely that the nets $n$ and $G(F(n))$ are equivalent, in the sense that they are subnets of each other, if we define subnet in the classical (Kelly) way. They're not identical, as they are defined on different index sets.

Long story short, we can translate all axioms for net convergence to ones for filterable convergence and vice versa.

The above 3 axioms for filters seem quite natural, and have net analogues. What Kelly considers is a characterisation (with extra axioms) of those convergence spaces that come from a topology. His axiom c) is a typical one that might fail for non-topological convergences.

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Thanks! (1) In Kelly's convergence class definition, are the nets all defined on a common directed set as their common index set, or their directed index sets vary among all possibilities? (2) Yes, Kelly's convergence class corresponds to a topology. Are some not all of the axioms in its definition equivalent to the definition of a convergence space defined in nlab? –  Tim Feb 23 '13 at 14:51
    
In Kelly's definition, all index sets are allowed. There is no common index set. But if we have two directed sets we can form their Cartesian product in the coordinatewise order, which is also directed, and then we can use projections to have a common directed set for the 2 nets, similarly as Kelly does for iterated limits. –  Henno Brandsma Feb 23 '13 at 15:06
    
No, the definitions as used for filters in the convergence space definition are pretty weak and do not characterise topological convergence as Kelly's does. There is a notion of convergence space for nets that only assumes the 2 first axioms (constant nets and subnets) and then we get a very broad class of (non-topological) convergence spaces. –  Henno Brandsma Feb 23 '13 at 15:15
    
Thanks! (1) Are you saying there is no way to generalize "convergence class" defined by Kelley, for example by omitting some of its axioms, to get an equivalence to "convergence space" defined in nlab? (2) Are there some axioms in Kelley's "convergence class" and nlab's "convergence space" equivalent to each other? –  Tim Feb 23 '13 at 15:24
    
As to (2): centred is equivalent to (a), the constant net. Isotone is equivalent to the subnet property. I don't see a direct analogue of the directedness among the net axioms, and we could translate (using the maps I described) the net axioms by Kelly to filters; try it. It has probably been done in the literature somewhere... –  Henno Brandsma Feb 23 '13 at 15:47
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