Sequences and closed sets.


by Fredrik
Tags: sequences, sets
Fredrik
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Jul26-11, 12:55 PM
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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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micromass
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Jul26-11, 01:04 PM
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Hi Frederik!

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.
Fredrik
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Jul26-11, 02:36 PM
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Quote Quote by micromass View Post
Hi Frederik!
Hi. I actually laughed out loud when I went back here after only ten minutes and saw that you had already replied. It's appreciated, as always. (I had to go out for a while after that. I would have replied sooner otherwise).

Quote Quote by micromass View Post
Every closed set of a topological space is closed under limits of sequences! It's the converse that's not true.
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.

Quote Quote by micromass View Post
[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.
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.


Quote Quote by micromass View Post
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.
Thanks. I wasn't familiar with this terminology.

micromass
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Jul26-11, 10:12 PM
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Sequences and closed sets.


Now that I think of it, your question would actually make an ideal exam question for my topology students So that's one less question I need to come up with. Thanks a lot!
wisvuze
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Jul26-11, 10:26 PM
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Hi micromass, if you remember us talking about topology books in the PF chatroom, this is discussed in the topology book by wilansky: http://www.amazon.com/Topology-Analy...1736932&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
micromass
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Jul26-11, 10:30 PM
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Quote Quote by wisvuze View Post
Hi micromass, if you remember us talking about topology books in the PF chatroom, this is discussed in the topology book by wilansky: http://www.amazon.com/Topology-Analy...1736932&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
It's too bad that I can't seem to find that book anywhere I've looked around for it, because I really want to read it. (I'm actually interested in the exercises)


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