# Countable intersection of open dense sets $\ne \emptyset$ implies Baire

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.

• Why did you delete your question shortly after getting an answer? May be you didn't know it, but we don't want you to do that on our site. This deprives the people, who put time and effort into answering, a chance to get appreciation from others. I'm not an expert on this topic, but superficially it looks like there is relatively little wrong with your question. Many members would like to see you explain a thing or two that you tried yourself. I refrain from judging your question in that sense. Commented Jan 12, 2015 at 17:57

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 $X$.
• @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 $q\in\Bbb Q\cap(0,1)$. Commented Sep 5, 2020 at 16:21