# Is there a name for a collection of open sets where arbitrary intersections are open?

Let $\mathcal{U} = \{U_i\}_{i\in I}$ be a collection of open sets with the property that the set $\bigcap_{i\in J} U_i$ is open for all subsets $J$ of $I$.

Is there a name for such collections of open sets?

Both locally finite collections and point-finite collections have this property, but these notions are too strong (just think of infinite discrete spaces).

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A topological space in which every collection of open sets has this property is called an Alexandrov space. –  Zhen Lin May 30 '12 at 9:27