I am facing the following definition of topology:
A topology on a set $X$ is defined as a subset $\cal O$ of the power set $\cal P$ of $X$, that:
- contains the empty set and $X$
- is stable over the arbitrary union
- is stable over the finite intersection
I was explained the last one in the following way: taken a finite number of subsets of $\cal O$, their intersection belongs to $\cal O$. However, the intersection of an infinite number of subsets of $\cal O$ may not belong to $\cal O$.
Honestly, this does not make sense to me. It seems to me that, taken infinite subsets of a set, their intersection by definition will belong to the set, so that the topology is inherently stable under infinite intersection.
Would you please explain me the concept of stability under finite intersection?