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1001:{*} 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.)
981:"mixture" can be understood like this.) This definition makes perfectly sense, if you want the map f:DβD' 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'
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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
945:
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
508:
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
980:
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
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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
946:
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.
928:. (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.) --
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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. --
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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
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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. --
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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.
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972:. 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
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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).
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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.
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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
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Right, so the confusing thing is the reference to being ordered by inclusion - the question being which direction.
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I am not familiar with nets. Is there a natural and important example of a subnet which is not monotone?
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Incidentally, Lang's book really isn't a good reference for this article. He uses the term
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is cofinal (the definition from the article) would be inappropriate. Indeed let us define
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is convergent to 0, so it would be inappropriate to consider this net a subset of
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put a short note that the discussion (if any) will continue here. --
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I see that I have caused that we discuss the same thing here and at
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only briefy in a discussion regarding completion of groups. --
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Ok, I agree that my wording was too strong -- I mean the word
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as a countable topological space, then the net (the sequence)
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