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June 17th, 2010, 09:21 AM   #1
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discrete topology

SUPPOSE that has the discrete topology .prove that is separable if and only if is countable.
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June 17th, 2010, 01:58 PM   #2
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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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