Posted by J. Holmer on September 5, 2007, 10:13 pm, in reply to "Separability" And your other comment is correct. You can replace "1" in the theorem we proved in class today with any fixed K >0. The statement then becomes: Suppose X is a metric space and there exists a number K>0 and an uncountable subset E such that for any x_1 \neq x_2 in E, \rho(x_1,x_2)\geq K. Then X is not separable.
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It is true that an uncountable set with the discrete metric is not separable. If a metric space has the discrete metric, then the only dense set is the whole space itself.
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