Under what circumstances is the discrete metric space separable? Can anyone help me please?
Under what circumstances is the discrete metric space separable?
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0I think this is an example to R. I'm not complaining, just answer the questions they ask – 2012-06-13
3 Answers
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.
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 $M$?
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.