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Is there a name for sets $S\in\mathfrak{A}$ for a complete lattice $\mathfrak{A}$ such that for all $T\subseteq \mathfrak A$, $$\bigvee T\in S \iff S\cap T\ne\varnothing?$$

Here $\bigvee$ is the join on the lattice $\mathfrak{A}$.

(Corrected: The name for $S$ not for $\mathfrak{A}$.)

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This is a very different question than you originally asked. – dfeuer Aug 27 '13 at 22:07

I don't know the required name, but your condition is involved in the definition of an open subset in Scott topology:

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