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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

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up vote 6 down vote accepted

Such collections have been called interior preserving, as in the definition of orthocompact space found here. An older and less descriptive term is Q-collection, as in this paper.

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Excellent! On the odd chance that somebody comes along with a better answer, I'll wait a little before accepting. –  Roar Stovner May 30 '12 at 9:59
    
@Roar: No problem: someone may have come up with a better name since I worked with them. –  Brian M. Scott May 30 '12 at 10:01

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