Infinite Intersections in Topology I'm having trouble finding an example that a professor wants. The problem is as follows.

Find an example of a topological space $(X,\mathcal{T})$ with open sets $U_{i}\in\mathcal{T}$, $i\in\mathbb{N}$, such that $\bigcap_{i=1}^{\infty}U_{i}\notin\mathcal{T}$.

I see why this intersection must not be in the topology from the definition alone. However, I'm not sure how to construct an example. Would the collection $A=\left\lbrace\left(-\frac{1}{n},\frac{1}{n}\right)\:|\:n\in\mathbb{N}\right\rbrace$ or the collection $B=\{(-m,m)\:|\:m\in\mathbb{N}\}$ constitute an example in $(\mathbb{R},\mathcal{T}_{std})$? 
Thanks in advance for any help!
 A: In the def'n of a topology, the intersection of $finitely$ many open sets is open. Example $A$ is correct, as $\cap A=\{0\}$ is not open in the usual topology. But $B$ is not an example, as $\cap B=(-1,1)$ is open.
A: Another example that I find interesting:  take $\mathbb{R}$ with the cofinite topology.  In this topology, a set is open if its complement is finite (plus, of course, we throw in the emptyset and all of $\mathbb{R}$, since those are required by the definition of a topology).  For example, the set $(-\infty,0)\cup (0,\infty)$ is open in this topology.  It is not to hard to check that this is, actually, a topology.
Now, let $U_n = \mathbb{R} \setminus \{n\}$, with $n$ ranging over the integers.  Each $U_n$ is open, but the intersection,
$$ \bigcap_{n\in\mathbb{Z}} U_n = \mathbb{R} \setminus \mathbb{Z}, $$
is not a cofinite set, and is therefore not open.
A: Taking complements, "find an infinite sequence of open sets whose intersection is not open" is equivalent to "find an infinite sequence of closed sets whose union is not closed." In the usual topology of the real line, all finite sets are closed, and so every countable set is the union of an infinite sequence of closed sets. Therefore, it will suffice to find a countable set of real numbers which is not closed in the usual topology; for instance, the set of all rational numbers, or the set of all reciprocals of positive integers.
