TITLE: Intersection of nested dense subsets.
QUESTION [9 upvotes]: Let $(A_n)$ be a decreasing sequence of subsets of the topological space $(X,\mathcal T)$ such that

$A_1$ is dense in $X$.
$A_{n+1}$ is dense in $A_n$.

is $\bigcap_{n=1}^\infty A_n$ dense in $X$?

REPLY [5 votes]: The best counterexample has already been given.
Let me simply point out that if your property holds for every such sequence $A_n$, then in particular $X$ would have to be a Baire space.
And there exist topological spaces which are not Baire spaces.
As you can see here, a typical counterexample is $\mathbb{Q}$ with the usual topology induced by the metric on $\mathbb{R}$.