Talk:Cofinal (mathematics)

Cofinal set of subsets

Is the set inclusion the right way round in the last substantive section? Isn't it supposed to read a is contained in b? That would make sense to me. Francis Davey 09:42, 9 August 2006 (UTC)

That would depend on the way the order relation on A ⊂ P(E) is defined: Given a,b ∈ A, a ≤ b could be defined as a ⊂ b or dually as b ⊂ a. I think this should be made clearer in the article. —Tobias Bergemann 10:34, 9 August 2006 (UTC)

Right, so the confusing thing is the reference to being ordered by inclusion - the question being which direction. Francis Davey 19:30, 9 August 2006 (UTC)

Cofinal function

In the current version of the article we have

A cofinal function is a function f: X → A with preordered codomain A such that its range f(X) is cofinal in the codomain.

When defining a subnet of a net in a topological space, a different notion of cofinal function is needed. (See e.g. Definition 3.3.14 in Runde: A Taste of Topology).

A function f:P→D from a preordered set to a directed set is cofinal if for each $d_{0}\in D$ there exists $p_{0}\in P$ such that $f(p)\geq d_{0}$ whenever $p\geq p_{0}$.

I do not have Lang's Algebra at me (which is the only reference mentioned in the article) at me. It is clear, that in case that f is monotone the two notions coincide. (This is the situation we see when dealing with the notion of cofinality for ordinals.) Do we have reference for the definition mentioned in the article? Are there applications where this notion is more appropriate than the definition I mentioned? See also Talk:Subnet (mathematics). --Kompik 22:22, 31 October 2007 (UTC)

Added: I checked it now, Lang's Algebra is given as a reference for a cofinal family of normal subgroups, i.e., it has nothing to do with the notion of cofinal map. --Kompik 20:27, 1 November 2007 (UTC)

I am not familiar with nets. Is there a natural and important example of a subnet which is not monotone? JRSpriggs 03:45, 1 November 2007 (UTC)

Well I don't think so. My opinion is that whatever can be done with a subnet given by a cofinal map, it can be done by a subnet given by a map which is cofinal and monotone. (I do not have a reference for this. But, for instance, proof of the theorem that a cluster point of a net is a limit of some subnet is often done in a such way, that the produced subnet is cofinal and monotone.) So I am not able to give such an example.

Maybe I should show why defining of subnet of $(x_{s})_{s\in S}$ as $(x_{f}(s'))_{s'\in S'}$ using only the requirement that the range of f:S→S' is cofinal (the definition from the article) would be inappropriate. Indeed let us define $f\colon \omega +\omega \to \omega$ given by $f(n)=n$ and $f(\omega +n)=0$. Clearly, f has cofinal range. If we consider $\omega +\omega$ with the usual ordering (as an ordinal) and $\omega$ as a countable topological space, then the net (the sequence) $(n)_{n<\omega }$ has no cluster point but the net $(f(a))_{a<\omega +\omega }$ is convergent to 0, so it would be inappropriate to consider this net a subset of $(n)_{n<\omega }$. (But that's not the example you asked for, it only shows, that "cofinal range" is not strong enough to define a good notion of a subnet - without adding some other property, as monotonicity or the condition above.) --Kompik 09:05, 1 November 2007 (UTC)

A different definition of cofinal map isn't really needed in the definition of a subnet. It is perfectly adequate to define a subnet via a monotone, cofinal map (i.e. one with a cofinal image), as in done in Willard. The definition used by Runde above really seems to mix the notion of cofinal image with a sort of "asymptotic monotonicity" that is strictly weaker than monotonicity. Also note that Runde's definition of cofinal only makes sense when the domain is a preordered set. To me the real question is whether or not there is a situation where Runde's definition is needed in place of a monotone, cofinal map.

Incidentally, Lang's book really isn't a good reference for this article. He uses the term cofinal only briefy in a discussion regarding completion of groups. -- Fropuff 21:36, 1 November 2007 (UTC)

Ok, I agree that my wording was too strong -- I mean the word needed. But with the exception of Willard's book, I have seen only Runde's definition before. Since then I have found it in several other books. At User:Kompik/Math/Subnet you can find comparing of definitions in various book -- Runde's definition seems to be older (traditional) one - e.g. Kelley, Engelking. I agree with the claim that both definitions are perfectly adequate in the sense that for both definitions we can prove results as: cluster point is a limit of some subnet, space is compact iaoi each net has a convergent subnet etc. However, I do not agree with the point that Runde's definition of cofinal map should be "tricky" or "unclear". (Your comment about "mixture" can be understood like this.) This definition makes perfectly sense, if you want the map f:D→D' to be convergent in some sense as a net. (To consider this map as a net, we need first define a topological space which contains D' in some reasonable way. This topological space can be defined on D' $\cup${*} in a similar way as one-point compactification of N can be identified with a convergent sequence. It would be necessary to add more details, I will stop here before getting it too long. The only point I want to stress is that this definition of cofinal map is not as meaningless as it might look at the first sight.)

I see that I have caused that we discuss the same thing here and at Talk:Subnet (mathematics). Perhaps we should agree on one place where to continue. I suggest to stay here and in Talk:Subnet (mathematics) put a short note that the discussion (if any) will continue here. --Kompik 12:05, 2 November 2007 (UTC)

I have changed the article on subnets in a such way that it uses the same definition of cofinal map as we have in this article. But I still do thing that it would be good to find a reference which mentions the definition cofinal map via the cofinality of the range for preordered or directed sets. (This approach is usual when dealing with ordinals, which are linearly ordered. Dealing with preordered sets can be slightly different.) As I have mentioned, the only book where I have seen the notion of cofinal map defined for directed sets was Runde. --Kompik 21:09, 10 November 2007 (UTC)