Under what circumstances is the discrete metric space separable? Under what circumstances is the discrete metric space separable?
Can anyone help me please?
 A: Recall that:


*

*If $A$ is a dense and closed subset of $X$ then $A=X$.

*In a discrete space every set is open, therefore every set is closed.

*If $X$ is discrete and separable then there is a countable subset which is dense.

A: A space $X$ is separable if it contains a dense countable subset $D$.
Now that we know the definition we need to think about what it means for $D$ to be dense in a discrete space. Dense means that if we pick any point $x$ in $X$ and an open set $O$ containing it, then $O$ will intersect with $D$. 
In a discrete space, the singleton set $\{x\}$ is open. The only way this set can have non-empty intersection with $D$ is if we have $x \in D$. 
But this means that the only dense subspace of a discrete space $X$ is $X$ itself. Hence, the only way to have a countable dense subset of a discrete space is if the space itself is countable.
A: Hint: Let $M$ be a metric space with the discrete metric.


*

*When is a subset $S\subseteq M$ closed?

*What is the closure of a set $S\subseteq M$?

*Thus, which subsets of $M$ are dense?

*When is there a countable dense subset of $M$?
