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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 $\nu(X) = 3$?

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    $\begingroup$ 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
    Commented Jul 25, 2017 at 11:17
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    $\begingroup$ Note that Taras' comment doesn't assume that the space is connected or Hausdorff. $\endgroup$
    – Ramiro de la Vega
    Commented Jul 25, 2017 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
    Commented Jul 25, 2017 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
    Commented Jul 25, 2017 at 16:28

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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.

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