Intersection of open sets in $\mathbb{R}$ 
Give examples of the following with justification. A collection of open subsets $(U_n)_{n\in\Bbb N}$ of $\Bbb R$ such that $\bigcap\limits_{n\in\Bbb N}U_n$ is not open.

as far as I'm aware the intersection of finitely many open sets is open, however, I'm assuming the fact I'm working in $\mathbb{R}$ is key in this question?
 A: It's not just $\Bbb R$, it's the fact that the space is $T_1$ without isolated points. This means that if $a$ is any point, then $R_a=\Bbb R\setminus\{a\}$ is a dense open set.
Now if $A$ is any set, then $A=\bigcap_{b\notin A}R_b$. In particular, if $A$ is not open, then this is an intersection of open sets which is not open again. And if $\Bbb R\setminus A$ is a countable set, it is a countable intersection.
You can probably try and find other conditions on $\Bbb R$ which imply this. Not just being $\Bbb R$ itself. Note, on the other hand, that in the discrete topology every intersection of open sets is an open set; as well in the trivial topology!
A: The intersection of finitely many open sets is open, this is part of the general definition of open sets. Infinitely many, though, is a different story. Take $U_n = \left(-\frac1n,\frac{1}{n}\right)$, for instance. The intersection of all of these is just $\{0\}$, which is not open.
There's nothing really special about $\Bbb R$ in this case. In some spaces you cannot find examples of such $U_n$, but in very many spaces you can.
A: $$\bigcap_{n\in\mathbb N^+}\left(-1-\frac1n,1+\frac1n\right)=[-1,1]$$
$$\bigcap_{n\in\mathbb N^+}\left(-\infty,\frac1n\right)=(-\infty,0]$$
Finitely many open sets must have open intersections. This is infinitely many sets.
