0
$\begingroup$

This is a special case of a question that has not been answered so far.

If $(X,\tau)$ is a connected $T_2$ space with more than 1 point, we define its nowhere dense covering number $\nu(X)$ by the smallest cardinality that a partition of $X$ into nowhere dense subsets of $X$ can have.

Question. Is there a connected Hausdorff space $(X,\tau)$ such that $\nu(X) = 3$?

$\endgroup$
  • 5
    $\begingroup$ Since the union of two nowhere dense sets is nowhere dense, the number $v(X)$ is always infinite. For meager spaces it is countable. But it cannot be equal to 3. $\endgroup$ – Taras Banakh Jul 25 '17 at 11:17
  • 3
    $\begingroup$ Note that Taras' comment doesn't assume that the space is connected or Hausdorff. $\endgroup$ – Ramiro de la Vega Jul 25 '17 at 11:44
  • $\begingroup$ BTW, when space $\ X\ $ has an isolated point than $\ \nu(X):=\infty\ $ by definition (it'd be only natural). $\endgroup$ – Wlod AA Jul 25 '17 at 11:51
  • $\begingroup$ Thanks @taras, can you post your comment here as an answer or in the more general question? $\endgroup$ – Dominic van der Zypen Jul 25 '17 at 16:28
2
$\begingroup$

Since the union of two nowhere dense sets is nowhere dense, the number $\nu(X)$ is always infinite. For meager spaces it is countable. But it cannot be equal to 3.

$\endgroup$

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.