# Countable Metric Space and nowhere dense sets

Hey I could not find an answer on wether or not a countable Union of nowhere dense sets is nowhere dense in a countable metric space. I know that a countable Union of nowhere dense sets is not always nowhere dense but I was wondering if the case were different in a countable metric space. If this is true or not true any explanation would be greatly appreciated.

• Hey, if every one-point set is nowhere dense, then every countable set is a countable union of nowhere dense sets. So, if you can find a countable metric space $X$ in which each one-point set is nowhere dense, but $X$ itself is not nowhere dense, that that's your counterexample. – bof May 10 '16 at 3:34
• Oh, ok that's makes sense. Thank you for clarifying! @bof – A.Riesen May 10 '16 at 3:42

In the rational numbers $\mathbb{Q}$, every set of the form $\{q\}$ is nowhere dense (it's not an isolated point!) and their (countable!) union is all of $\mathbb{Q}$, so the opposite of nowhere dense (everywhere dense) in $\mathbb{Q}$.