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 June 17th, 2010, 09:21 AM #1 Newbie   Joined: Jun 2010 Posts: 2 Thanks: 0 discrete topology SUPPOSE that $X$ has the discrete topology .prove that $X$ is separable if and only if $X$ is countable.
 June 17th, 2010, 01:58 PM #2 Senior Member   Joined: Feb 2009 Posts: 172 Thanks: 5 Re: discrete topology Consider $X$ with the discrete topology. $(\Rightarrow)$ Suppose $X$ is separable. Then there exists $Y\subset X$ wich is coutable and dense. Since $X$ is discrete $Y=\overline{Y}$ for all $Y\subset X$. Hence, the only dense subset of $X$ is $X$ and so $X$ must be countable. $(\Leftarrow)$ Suppose $X$ is coutable. Since $X$ is dense on itself we have that $X$ is separable.

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