Intersection of nested dense subsets.

Question

Let $(A_n)$ be a decreasing sequence of subsets of the topological space $(X,\mathcal T)$ such that

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

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

Answer 1

Hint: Take $\{r_n,n\in\Bbb N\}$ an enumeration of rationals and $X_n:=\{r_k,k\geqslant n\}$. Each $X_n$ is dense in $X:=\Bbb R$ for the usual topology, but the intersection is empty.

Answer 2

HINT: Let $\{q_n:n\in\Bbb Z^+\}$ be an enumeration of $\Bbb Q$. For $n\in\Bbb Z^+$ let $A_n=\{q_k:k\ge n\}$. What is $\bigcap_{n\ge 1}A_n$?

Answer 3

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 $\mathbb{R}$.