1
$\begingroup$

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.

$\endgroup$
  • 3
    $\begingroup$ 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. $\endgroup$ – bof May 10 '16 at 3:34
  • $\begingroup$ Oh, ok that's makes sense. Thank you for clarifying! @bof $\endgroup$ – A.Riesen May 10 '16 at 3:42
1
$\begingroup$

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 $\mathbb{Q}$.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.