In which topologies do open sets maintain open under countable or arbitrary intersection?

Question

We know that in the usual topology, countable or arbitrary intersection of open sets can zoom into a singleton, hence is not in the topology.

I am curious if there is well known classes of topologies, whose open sets maintains their openness after countable or arbitrary intersection.

I know two examples:

1) In the discrete topology, $\mathcal{T} = \mathcal{P}(X)$. The power set is a $\sigma$-algebra, hence "closed" under countable intersections, so a countable intersection of open sets in the discrete topology is open.

2) In the particular point topology, given $x_o \in \mathbb{R}$, $\mathcal{T}_{x_o} = \{U \subseteq \mathbb{R}| x_o \in U\}\cup\{\varnothing\}$, an arbitrary intersection of open sets will always contain $x_o$, in particular $x_o$ is open. Hence a countable intersection of open sets in the particular point topology topology is open.

Are there a lot more well known classes of topology whose condition on intersections of open set can be extended?

Answer 1

Spaces in which countable intersections of open sets are open are called $P$-spaces. (Warning: the same term is also used with a completely different meaning.) The co-countable topology on an uncountable set is an example of a non-discrete $T_1$ $P$-space. In general we can start with any space $\langle X,\tau\rangle$ and let $\tau'$ be the collection of $G_\delta$ sets with respect to $\tau$: $\langle X,\tau'\rangle$ is then a $P$-space.

Spaces in which arbitrary intersections of open sets are open are called Alexandrov spaces; they are precisely the spaces in which each point has a smallest open nbhd, which is of course equal to the intersection of all nbhds of the point. As the Wikipedia article explains, all Alexandrov spaces arise in the following way. Start with any preorder $\preceq$ on $X$. For each $x\in X$ let $B_x=\{y\in X:x\preceq y\}$, and let $\mathscr{B}=\{B_x:x\in X\}$; then $\mathscr{B}$ is a base for an Alexandrov topology on $X$, and for each $x\in X$, $B_x$ is the smallest open nbhd of $x$.