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I know that $[0,1]^{\Bbb R}$ with the product topology is not 1st countable. What I want now is to find a subset of $[0,1]^{\Bbb R}$ which is not closed but has all limit points. Does such a set exist? Then, what is it?

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    You should define what you mean by "limit point" here; the usual definition is that $x$ is a limit point of $A$ if every neighborhood of $x$ contains a point of $A$ (other than $x$ itself). With this definition, in *any* topological space, a set is closed iff it contains all its limit points. So I suppose you want "limit point" to mean something else.2012-10-06

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