June 17th, 2010, 09:21 AM  #1 
Newbie Joined: Jun 2010 Posts: 2 Thanks: 0  discrete topology
SUPPOSE that has the discrete topology .prove that is separable if and only if is countable.

June 17th, 2010, 01:58 PM  #2 
Senior Member Joined: Feb 2009 Posts: 172 Thanks: 5  Re: discrete topology
Consider with the discrete topology. Suppose is separable. Then there exists wich is coutable and dense. Since is discrete for all . Hence, the only dense subset of is and so must be countable. Suppose is coutable. Since is dense on itself we have that is separable. 

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