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Oops, you're right. In the original statement I spoke of limit points of S in S, which worked when it was empty, but not currently. I added the statement about infinite subsets having limit points because this is the definition of limit point compactness (in some texts), and is an important property
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The second is not correct. Given a singleton S={s}, s is not a limit point of S, because it does not lie in the closure of S-{s}, which is empty. It can still be generalized somewhat, however. I changed it to this, maybe the language could be cleaned up some:
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Given a subset S of X which is not a singleton, all elements of X are limit points of S. If S is a singleton, every point of X \ S is still a limit point of S. In particular, every infinite subset of X has a limit point.
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Yup you're right, except that your version doesn't work when S is empty. I also don't see the point of the last statement about infinite sets: you don't need infinitely many elements, you only need two.
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of metric compact spaces. If there were a page on limit point compactness I'd link to that instead.
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To AxelBoldt: most of your changes were good, but you changed this line:
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