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micromass
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#2
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.