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I am thinking about this question for a while. Is any infinite set $X$ on the line separable? I think the answer is yes, but to do so, I need to show there exists a countable subset $A$ in $X$ such that $X$ is in the closure of $A$.

Can anyone outline the proof for me? Thanks.

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    By "the line", do you mean the "real line". And do you put on this set $X$ the topology given by $\mathbb R$?2011-12-07
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    While it is not *very* relevant, it may still be of some interest to people reading this thread: without the axiom of choice it is consistent that there is a subset of real numbers which is not separable.2011-12-08
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    @Asaf: in my opinion your comment is totally irrelevant but somewhat interesting. :)2011-12-08

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