If $X$ is a baire space, then every countable interesction of open dense sets is dense in $X$.

Now, assume that $X$ is some topological space with the propery:

If $A_{1},A_{2},...$ are open dense sets in $X$, then $\bigcap _{n=1} ^{\infty} A_{n} \ne \emptyset$.

I'm looking for space $X$ which respects that property, but is not a Baire space. In other words, i'm looking for a space $X$ such that every countable intersection of open dense sets is non-empty, but not necessary dense.

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1 Answer 1


Let $X$ be the disjoint union of $\Bbb Q$ and an isolated point. Every countable intersection of dense open sets must contain the isolated point, but its singleton is not dense and is the intersection of a countable family of dense open sets.

Equivalently, let $X=\big(\Bbb Q\cap(0,1)\big)\cup\{2\}$ as a subspace of $\Bbb R$.

A slightly less trivial example is $X=\big(\Bbb Q\cap(0,1)\big)\cup[2,3]$ as a subspace of $\Bbb R$: every countable intersection of dense open sets will be dense in $[2,3]$ but need not be dense in $X$.

  • $\begingroup$ In the 2nd example, how can we deduce that the intersection of any countable collection of dense-open sets must be equal to {2}? $\endgroup$
    – Saikat
    Sep 5, 2020 at 5:44
  • $\begingroup$ @SaikatGoswami: That isn’t true. What is true is that there is some countable family of dense open sets whose intersection is $\{2\}$, namely, the family of sets $X\setminus\{q\}$ for $q\in\Bbb Q\cap(0,1)$. $\endgroup$ Sep 5, 2020 at 16:21

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