I studying topology by reading the “Topology without Tears” written by Sidney Morris. One can easily find the book in pdf.
I know that the intersection of open sets is not always an open set. However, I was trying to find topologies that verify that. I came up with $\{ \emptyset, \mathbb{N} \}$ and all the sets whose elements are only powers of 2.
Can you give some other examples and what conditions such topologies must have to satisfy this property?
On
Spaces in which any intersection of open sets is open are called Alexandrov spaces. You will find several characterizations of these spaces if you follow this link, in particular their connections with preorders via the specialization preorder.
On
OK so the real numbers with the usual topology also show this. Let us think about the intersection of all open sets $(-\epsilon, \epsilon)$ where $\epsilon$ can be any real number. This intersection must be $\{0\}$ because let us say that some $x \neq 0$ is in there then we just need to choose $\epsilon$ with magnitude less than $x$, $x$ will not be in $(-\epsilon, \epsilon)$ so $x$ could not have been in the intersection. Of course single element sets in the reals are closed and not open so $\{0\}$ is not open.
Cleary there is nothing special about $0$ in this example i.e. we could pick the intersection of all the intervals $(y-\epsilon, y+\epsilon)$ and this would just be $\{y\}$
The only Hausdorff (or even $T_1$) topological spaces in which intersections of open set are always open are the discrete ones. Any set is a union of singletons (which are closed) and so any set is an intersection of open sets. So every set is open.