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Sequences and closed sets. 
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#1
Jul2611, 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?



#2
Jul2611, 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 wellknown, all firstcountable spaces are sequential. 


#3
Jul2611, 02:36 PM

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#4
Jul2611, 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!



#5
Jul2611, 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/TopologyAnaly...1736932&sr=81
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 


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