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I know that every separable metric space has a countable base.

I was wondering if we can get a countable dense subset from every metric space that has a countable base.

Thank you very much!!

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You don't even need a metric space for that direction, in holds in general. If your topology has a countable base, just pick one element out of every base set and you will get a countable dense subset.

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  • $\begingroup$ Great, thank you very much! $\endgroup$ – julian.marr Jun 24 '15 at 22:17
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In general in any topological space let $A=\{a_i : a_i \in O_i\}$ you can prove that $A$ is a dense countable subset, such that $B=\{O_i : i \in\mathbb N\}$ is the Base of topological space

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  • $\begingroup$ You are welcome $\endgroup$ – Upgrade Jun 24 '15 at 22:17
  • $\begingroup$ Not every topological space has a countable basis. $\endgroup$ – Asaf Karagila Jul 21 '15 at 7:36

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