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Under what circumstances is the discrete metric space separable? Can anyone help me please?

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    I think this is an example to R. I'm not complaining, just answer the questions they ask2012-06-13

3 Answers 3

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Recall that:

  1. If $A$ is a dense and closed subset of $X$ then $A=X$.
  2. In a discrete space every set is open, therefore every set is closed.
  3. If $X$ is discrete and separable then there is a countable subset which is dense.
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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 $M$?

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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.