Infinite intersection of open sets need not be open The following is the property of an open set:
The intersection of a finite number of open sets is open.
Why is it a finite number? Why can't it be infinite?

Let $U_n=(-1/n,1/n)$ for $n=1,2,3,\dots$.  Then $U_n$ is open for all $n$. Suppose $x\in\cap_n U_n$.  Then if $x>0$ there is an $n$ such that $\frac1n<x$ (since $\frac1n\to0$).  Thus $x\not\in U_n$ for that $n$.   Same if $x<0$. Thus $x\not\in\cap_n U_n$.  Now $0\in U_n$ for all $n$. Thus $\cap_n U_n=\{0\}$ which is not open.   
