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Let M be an uncountable discrete metric space. Then M is

 A) separable B) not separable C) empty D) none of these

 A) separable

Let $$A\subset M$$ and $$A\ne M$$

Since any set is closed in the discrete metric space M

$$\Rightarrow \overline{A}=A$$

$$\therefore$$ A is not dense.

Hence any uncountable discrete metric space is not seprable.

Part of solved Real Analysis questions and answers : >> Elementary Mathematics >> Real Analysis

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