An arbitrary union, or a finite intersection, of open sets in a topological space is again open. What name is given to the hypothetical property that an *arbitrary* intersection of open sets is open?


As an example, consider a partially ordered set $X$. Call a subset $U\subseteq X$ open if $y\le x\in U$ implies $y\in U$. (Bonus question: Are there other interesting examples?)



What do you call a topology that is closed under arbitrary intersections?
[Alexandrov spaces](http://en.wikipedia.org/wiki/Alexandrov_topology).


