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Sequences and closed sets.

  1. Jul 26, 2011 #1

    Fredrik

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    Anyone have a good example of a closed subset of a topological space that isn't closed under limits of sequences?
     
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  3. Jul 26, 2011 #2

    micromass

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    Hi Frederik! :smile:

    Every closed set of a topological space is closed under limits of sequences! It's the converse that's not true. That is: there are sets which are not closed but which are still closed under limits of sequences.

    For example, take the cocountable topology. Let X be a set and set

    [tex]\mathcal{T}=\{A\subseteq X~\vert~X\setminus A~\text{is countable}\}\cup\{\emptyset\}[/tex]

    Every convergent sequence in this topology is (eventually) a constant sequence. Thus all sets are closed under limits of sequences. But not all sets are closed, of course.

    Some terminology: a set that is closed under limits of sequences is called sequentially closed. A topological space where closed is equivalent with sequentially closed, is called a sequential space. As is well-known, all first-countable spaces are sequential.
     
  4. Jul 26, 2011 #3

    Fredrik

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    Hi. I actually laughed out loud when I went back here after only ten minutes and saw that you had already replied. :smile: It's appreciated, as always. (I had to go out for a while after that. I would have replied sooner otherwise).

    Ah yes. I actually had that right in my mind a few minutes earlier, but somehow got it wrong anyway when I made the post. This is what I was thinking before my IQ suddenly dropped 50 points: In a metric space, a set is closed if and only if it's closed under limits of sequences. In a topological space, the corresponding statement is that a set is closed if and only if it's closed under limits of nets. Since sequences are nets, a closed set must be closed under limits of sequences. These statements suggest that there's a set E that's closed under limits of sequences and still isn't closed. Then there should exist a convergent net in E, that converges to a point in Ec. That's the sort of thing I originally meant to ask for an example of, but your example illustrates the point as well.

    It took me a while to understand this, but I get it now. It's a good example. It's a weird topology since even 1/n→0 is false in this topology. I think I also see an example of the kind I originally had in mind: Consider the cocountable topology on ℝ. Let E be the set of positive real numbers. Let I be the set of all open neighborhoods of 0 that have a non-empty intersection with E. Let the preorder on I be reverse inclusion. For each i in I, choose xi in i. This defines a net in E with limit 0, which is not a member of E.


    Thanks. I wasn't familiar with this terminology.
     
  5. Jul 26, 2011 #4

    micromass

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    Now that I think of it, your question would actually make an ideal exam question for my topology students :biggrin: So that's one less question I need to come up with. Thanks a lot!
     
  6. Jul 26, 2011 #5
    Hi micromass, if you remember us talking about topology books in the PF chatroom, this is discussed in the topology book by wilansky: https://www.amazon.com/Topology-Ana...9034/ref=sr_1_1?ie=UTF8&qid=1311736932&sr=8-1

    and the exact same answer/example is given too, with the cocountable topology and how every sequence would have to be eventually constant. ( it's cool! )
    Not that I'm contributing much to the conversation, but I just wanted to point that out
     
  7. Jul 26, 2011 #6

    micromass

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    It's too bad that I can't seem to find that book anywhere :frown: I've looked around for it, because I really want to read it. (I'm actually interested in the exercises)
     
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