Why in a Hausdorff sequentially compact space the size of the closure of a countable subset is less or equal than $c$ ? I can see why this is true when the space if first countable but we are not assuming so.
Size of the closure of a set
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general-topology
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2@Bogdan: How do you know it's true? Also where is this question from? I can't decide if it's true or not. But none of the counterexamples in 'Counterexamples in Topology' will work. – 2010-11-23
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4I posted an answer pointing out that the Stone–Čech compactification of $\mathbb{N}$ is compact Hausdorff of size $2^{2^{\aleph_0}}$ and has a countable dense set. But it seems not to be sequentially compact, so it doesn't quite make a counterexample. – 2010-11-24
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2I found http://at.yorku.ca/cgi-bin/bbqa?forum=ask_a_topologist;task=show_msg;msg=1381.0001 but it says that any point in the closure is a sequential limit, which is not obvious to me in the absence of first-countability. It may be that this follows from sequential compactness but I don't see how. – 2010-11-24
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1@Nate: I found that too and agree with you that it isn't obvious. Also, thanks for asking for a clarification there. – 2010-11-24
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2@Nate Eldredge: Do you think this question is inappropriate to MathOverflow? Maybe it's not true at all, and a counterexample doesn't seem easy to find. – 2010-11-27
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1@Nuno: Seems like a perfectly reasonable MO question to me, and a recent meta.MO thread seemed to agree that it is fine to repost math.SE questions to MO (where appropriate for the latter) after a reasonable time with no answers. – 2010-11-27
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1@Bogdan: I think you should post this question on MathOverflow, since you didn't get any right response here. If you don't mind, I can post it there for you. @Nate Eldredge: Good to know that. Do you have the link for the thread? I search a bit, but didn't find it. Maybe because I'm not familiar with meta.MO. Also, if Bogdan don't answer this comment we can post this question there. – 2010-11-27
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0@Nuno: http://tea.mathoverflow.net/discussion/791/what-is-a-duplicate/#Item_0 – 2010-11-28
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0@Nate Eldredge: Thank you very much. – 2010-11-28
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0One more comment: the claim at that "For pseudoradial Hausdorff spaces X we have |X| <= d(X)^c(X) (<= 2^d(X))" (at the at.yorku.ca link) is not correct (it is the second inequality that is a problem.) Otherwise we would have a fine contradiction coming from the fact that if $\mathfrak{c}\leq\aleph_2$ then all compact, sequentially compact spaces are pseudo-radial, but my (consistent) counterexample is a separable, compact, sequentially-compact space (hence pseudo-radial) and $\mathfrak{c}=\aleph_2$ but $|X|>2^{\aleph_0}$. – 2011-02-08